Hi, all,

A lot of recent discussions involving uses of commas left me realizing that

there is a whole bunch that I don't know. I've almost replied and asked

questions a dozen or so times recently, but ultimately I was too confused to

even come up with a question. So now I'm betting back to the latent realm

of questioning...

I took a scan through the tuning dictionary and I really didn't find what I

was hoping to see, which is kind of a summary of all the different ways that

commas can be used.

(Gee Monz, its not your fault -- its just that all my desires for knowledge

now get projected onto the tuning dictionary.)

Commas can be "left in" (not necessarily a piece of terminology), "tempered

out", "distributed" and perhaps many other possibilities. It would be nice

to see all these possibilities explained in one place. This is closely tied

in with unison vector and tempering, but actually *tempering* is not

necessarily necessary, and furthermore temperament sometime happens when it

is not intended, as per a recent conversation with Carl in which he informed

me that even by just playing an approximate 7-limit interval that occurs in

a 5-limit scale I have invoked "temperament". Maybe its too much to ask but

I'd love to see this all tied together in one comprehensive tutorial!

It seemed to me also that in the process of defining an actual tuning that

another concept of comma is perhaps introduced, but according to Carl this

isn't called a comma. When distributing a comma, which I think implies a

unison vector and also implies that the comma is rational (whereas the

historical use does not imply that, right?), the mechanics of distributing

it are likely (but not guaranteed I don't think) to introduce irrational

distributions among the various dimensions of the lattice. This

distributing reminds me of commas again, smaller commas, and my original

nieve intepretation of commas in temperament involved this perhaps

misfounded concept. I'm pretty vague on this, and welcome feedback.

The tonalsoft comma page currently relies heavily on a quote from Dave

Keenan which gives two definitions of comma. However neither definition

seems to say much about the functions of commas as commonly discussed on

this list, nor is any restriction to rational values mentioned. It seems to

me these comma functions apply regardless of the size of the comma, even if

they may seem a little absurd in some cases. So I'd almost wish there was

another word for this use of "comma", and it doesn't quite look like

"anomaly" is that word. But we could also go on with this usage and when

necessary invoke some term like "generalized comma" to indicate that a comma

has gotten outside of its normal size range, and perhaps "functional comma"

to indicate that it is the function and not the size that is at stake.

Around here everyone will usually know this from context. Nonetheless for

interfacing with a larger world we may need some kind of "qualification"

terms which can be invoked when necessary, probably rarely.

Has Paul dealt with this in his recent writings? I haven't had the time to

look at any of it yet.

Just doing my newbie job. ;)

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> I took a scan through the tuning dictionary and I really didn't

find what I

> was hoping to see, which is kind of a summary of all the different

ways that

> commas can be used.

...

> Commas can be "left in" (not necessarily a piece of

terminology), "tempered

> out", "distributed" and perhaps many other possibilities.

"Tempered out", "distributed", and "made to vanish" are all the same

thing in regard to commas (and schismas, kleismas, dieses etc. if we

disallow the generic use of "comma").

By the way "anomaly" doesn't work for me as a substitute for this

generic use of "comma" because of its implication of "a failure to

meet expectations".

If a comma (generic sense) is not tempered out (tempered to zero),

then tempering may make it larger, smaller or even negative,

relative to its untempered size, or it may remain untempered.

If it is "left in" then it may be an actual scale step (not likely,

but more likely for dieses and larger), or it may be a chromatic

alteration of a scale notes when modulation occurs, or it may be

symbolised as an accidental and used to notate scale notes even when

the note without that accidental never occurs. Or it may not

explicitly appear at all.

There is a borderline case where although there is no tempering, a

comma may already be so small that it can be ignored. So although it

vanishes it isn't exactly _made_ to vanish. This is most likely for

kleismas and schismas and smaller.

> It seemed to me also that in the process of defining an actual

tuning that

> another concept of comma is perhaps introduced, but according to

Carl this

> isn't called a comma. When distributing a comma, which I think

implies a

> unison vector and also implies that the comma is rational (whereas

the

> historical use does not imply that, right?),

Commas are always rational, ... until they are tempered. Hmm. That

doesn't sound too clear. :-) A comma is always _defined_ as a

rational frequency ratio, or equivalently as a prime-exponent-

vector. You can't just pick any random number of cents and call it a

comma.

> the mechanics of distributing

> it are likely (but not guaranteed I don't think) to introduce

irrational

> distributions among the various dimensions of the lattice.

Correct.

> This

> distributing reminds me of commas again, smaller commas, and my

original

> nieve intepretation of commas in temperament involved this perhaps

> misfounded concept. I'm pretty vague on this, and welcome

feedback.

>

You're right. It is misfounded. The quarter-comma of quarter-comma

meantone, is not itself a comma, because it doesn't arise as the

difference between two different stacks of rational intervals, and

cannot be defined in rational terms. The fourth root of 81/80 is

irrational.

> The tonalsoft comma page currently relies heavily on a quote from

Dave

> Keenan which gives two definitions of comma. However neither

definition

> seems to say much about the functions of commas as commonly

discussed on

> this list,

No, nothing about functions, sorry.

> nor is any restriction to rational values mentioned.

Technically, you're right! Yikes! But it was certainly my intention

to so restrict it.

Monz, if you could change those two ocurrences of "pitch ratios"

to "rational pitches" that should solve the problem.

> It seems to

> me these comma functions apply regardless of the size of the

comma, even if

> they may seem a little absurd in some cases.

Agreed. But I did include the deliberately vague

proscription "typically smaller than a scale step".

> So I'd almost wish there was

> another word for this use of "comma",

Me too. One suggestion I have is "komma".

But whether using "komma" or "comma" to cover this broader range of

sizes that have the same range of functions, it is well to spell out

the fact that you are doing so.

> and it doesn't quite look like

> "anomaly" is that word.

Agreed.

> But we could also go on with this usage and when

> necessary invoke some term like "generalized comma" to indicate

that a comma

> has gotten outside of its normal size range, and

perhaps "functional comma"

> to indicate that it is the function and not the size that is at

stake.

Those sound reasonable to me. The biggest disagreement is the one

about what functions a comma may serve and still be a comma (or

still be commatic). I haven't yet seen anything that would make me

accept that "chromatic comma" (non-vanishing comma) is some kind of

oxymoron.

> Around here everyone will usually know this from context.

Nonetheless for

> interfacing with a larger world we may need some kind

of "qualification"

> terms which can be invoked when necessary, probably rarely.

>

> Has Paul dealt with this in his recent writings? I haven't had

the time to

> look at any of it yet.

Paul Erlich has followed Paul Hahn in using "commatic" as the

opposite of "chromatic". The _sound_ of these two words is certainly

seductive in making one want them as opposites. "chromatic"

versus "vanishing" certainly doesn't sound as nice. And the other

seductive fact is that for about the last 400 years in the west, the

most important comma, the syntonic comma, has been something that

vanishes, and for about the last 100 years, the second most

important comma, the Pythagorean comma, has also been something that

vanishes.

But the fact is that both of these were originally named as commas

at a time when temperament was unknown.

> Just doing my newbie job. ;)

And a great job it is!

Hi Kurt,

>Maybe its too much to ask but

>I'd love to see this all tied together in one comprehensive tutorial!

Have you read Paul's papers?

>It seemed to me also that in the process of defining an actual tuning that

>another concept of comma is perhaps introduced, but according to Carl this

>isn't called a comma.

As you can see from the recent threads here, there is anything but

agreement on what a "comma" is. Nevertheless, the language seems to

work among those who already understand the concepts.

>When distributing a comma, which I think implies a

>unison vector

As it happens, there's also very little agreement on what a

"unison vector" is.

>and also implies that the comma is rational (whereas the

>historical use does not imply that, right?),

It may be said that commas are *usually* rational.

>The tonalsoft comma page currently relies heavily on a quote from Dave

>Keenan which gives two definitions of comma. However neither definition

>seems to say much about the functions of commas as commonly discussed on

>this list, nor is any restriction to rational values mentioned.

It's clear we've a long way to a consensus on this stuff, let alone

a clear tutorial on it. My advice is to ignore terminology, and

instead start working out tunings that interest you. In the process

you will aquire the tools you need.

-Carl

hi Kurt and Dave,

first i want to say: Kurt, that is one of the

greatest subject-lines i've ever seen around here. :)

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

>

> > I took a scan through the tuning dictionary and I

> > really didn't find what I was hoping to see, which is

> > kind of a summary of all the different ways that

> > commas can be used.

> ...

> > Commas can be "left in" (not necessarily a piece of

> > terminology), "tempered out", "distributed" and perhaps

> > many other possibilities.

>

> "Tempered out", "distributed", and "made to vanish" are

> all the same thing in regard to commas (and schismas,

> kleismas, dieses etc. if we disallow the generic use of

> "comma").

i should say now that the "temper" page in the Encyclopaedia

http://tonalsoft.com/enc/index2.htm?temper.htm

which includes "temper out (vanish)" as its "part 2",

was very quickly thrown together, just so that i had

a page that defined the concept. there is a *lot*

(and i mean *really* a lot) more that i can add there.

i try to make a page for everything that i think

should be there, but even so, there are still some

very important terms that have not made it into

the Encyclopaedia. "flat" and "sharp" only just

got their pages a few weeks ago (5 years after the

inception of the Dictionary) -- and those two terms

are both pretty darn basic to tuning!

guess i was just too busy dealing with "periodicity-block"

and "unison-vector" and "val" and "proslambanomonos" ...

;-)

> By the way "anomaly" doesn't work for me as a substitute

> for this generic use of "comma" because of its implication

> of "a failure to meet expectations".

i never really liked "anomaly" either ... i got it

from Wilkinson, and he does indeed use it to describe

small JI intervals which fail to close the system,

which i suppose is an EDO-centric viewpoint and

therefore really shouldn't be extended to JI.

but i really don't like the use of "comma" in its

more general sense, precisely *because* it has such

a long history of being used to designate an interval

of a particular size-range.

"diesis" has been already suffered great variability

in meaning over the course of its history, and now

*we* are doing the same thing to "comma".

i would much like to have a general term to refer

to *all* of these small intervals -- schisminas,

skhismas, kleismas, commas, dieses, and whatever else.

but let's come up with a new one, not create a

secondary definition for "comma" (well, i guess that's

already a done deal ... but i for one will not advocate

its use with the general meaning).

if "anomaly" is unacceptable or unpopular, fine

-- so let's think up something else that's good.

> There is a borderline case where although there is no tempering,

> a comma may already be so small that it can be ignored. So

> although it vanishes it isn't exactly _made_ to vanish. This

> is most likely for kleismas and schismas and smaller.

and that is *precisely* what Fokker meant by "unison-vector".

so that is the term which should be used to designate that case.

> > It seemed to me also that in the process of defining

> > an actual tuning that another concept of comma is perhaps

> > introduced, but according to Carl this isn't called a comma.

> > When distributing a comma, which I think implies a

> > unison vector and also implies that the comma is rational

> > (whereas the historical use does not imply that, right?),

>

> Commas are always rational, ... until they are tempered.

> Hmm. That doesn't sound too clear. :-) A comma is always

> _defined_ as a rational frequency ratio, or equivalently

> as a prime-exponent-vector. You can't just pick any random

> number of cents and call it a comma.

Dave, your last sentence is true. but -- and i'm assuming

you're using "comma" in the general sense here -- you

certainly can find a "comma" which exists between pitches

at the two extremes of a meantone chain. those two pitches

have irrational "ratios", and so does that "comma".

for example, the interval between the origin and the

8ve-reduced 31st generator of 1/4-comma meantone, is

2^18 * 5^-(31/4) = ~ 6.068717548 cents. that "ratio"

is definitely an irrational number ... and in fact,

is the interval which vanishes in 31edo, and also

which is *why* 31edo is such a good emulation of

1/4-comma meantone.

> > The tonalsoft comma page currently relies heavily on

> > a quote from Dave Keenan which gives two definitions of

> > comma. However neither definition seems to say much

> > about the functions of commas as commonly discussed on

> > this list,

>

> No, nothing about functions, sorry.

>

> > nor is any restriction to rational values mentioned.

>

> Technically, you're right! Yikes! But it was certainly my

> intention to so restrict it.

>

> Monz, if you could change those two ocurrences of

> "pitch ratios" to "rational pitches" that should solve

> the problem.

well ... i don't think that's a good idea, as per my

discussion above.

we need to discuss this more before i change that

definition. perhaps it's a good idea to have separate

terms to differentiate between rational "commas" and

irrational ones.

(there i go, encouraging the proliferation of jargon again ...)

> > It seems to me these comma functions apply regardless

> > of the size of the comma, even if they may seem a

> > little absurd in some cases.

>

> Agreed. But I did include the deliberately vague

> proscription "typically smaller than a scale step".

>

> > So I'd almost wish there was

> > another word for this use of "comma",

>

> Me too. One suggestion I have is "komma".

>

> But whether using "komma" or "comma" to cover this

> broader range of sizes that have the same range of

> functions, it is well to spell out the fact that you

> are doing so.

if you recall, i too at first accepted the komma/comma

difference in spelling to represent the difference

in meaning ... but gave it up in the face of protest

from others ... and i largely agreed with them,

because of the similarity of spelling.

a totally new word would be better, and i will

continue to maintain that position in the face

of all contrary arguments.

> > and it doesn't quite look like

> > "anomaly" is that word.

>

> Agreed.

as i said, i don't really care for "anomaly" either

... but it *does* get across the meaning of a small

rational interval which almost-but-not-quite closes

the tuning into a circle.

> > But we could also go on with this usage and when

> > necessary invoke some term like "generalized comma"

> > to indicate that a comma has gotten outside of its

> > normal size range, and perhaps "functional comma"

> > to indicate that it is the function and not the size

> > that is at stake.

>

> Those sound reasonable to me.

i still have to disagree. IMO, it's always better

to have a single short word to represent a concept

instead of a more general word with a preceding

qualifier which narrows its meaning.

i'll say it again: new terms, however strange they

seem at first, become familiar with usage!

every language that has enough speakers/writers to

keep it alive will always keep growing. new words

are invented all the time ... what really good

argument can anyone present against that? sorry,

i just won't accept the "too hard for newbies" line.

my Encyclopaedia is never further than a mouse-click

away, and if the word isn't in there or isn't well-defined,

then it should be and eventually will be.

> The biggest disagreement

> is the one about what functions a comma may serve and

> still be a comma (or still be commatic). I haven't yet

> seen anything that would make me accept that "chromatic comma"

> (non-vanishing comma) is some kind of oxymoron.

the biggest problem i've always had with "chromatic

unison-vector" or "chromatic comma" is that the Greek

root "chromatic" refers to scale elements which are

approximately a semitone apart, and i don't want to

see that "chromatic" referring to intervals which are

much smaller than that -- which *is* very possible

according to the recent usages i've seen.

and as i've said, the Greek root "chroma" already

has a significantly different meaning (essentially

the same as "pitch-class") ... so this could get messy,

but i will do everything i can to prevent that.

> > Around here everyone will usually know this from

> > context. Nonetheless for interfacing with a larger

> > world we may need some kind of "qualification"

> > terms which can be invoked when necessary, probably

> > rarely.

> >

> > Has Paul dealt with this in his recent writings?

> > I haven't had the time to look at any of it yet.

>

> Paul Erlich has followed Paul Hahn in using "commatic" as

> the opposite of "chromatic". The _sound_ of these two words

> is certainly seductive in making one want them as opposites.

> "chromatic" versus "vanishing" certainly doesn't sound as nice.

> And the other seductive fact is that for about the last

> 400 years in the west, the most important comma, the

> syntonic comma, has been something that vanishes, and for

> about the last 100 years, the second most important comma,

> the Pythagorean comma, has also been something that

> vanishes.

>

> But the fact is that both of these were originally named

> as commas at a time when temperament was unknown.

which strengthens my argument that we really need a

term to designate the "vanishingness" of a small interval.

i propose "vapro", for "VAnishing PROmo".

and i also like the fact that it brings to mind "vapor",

which is sort of like what those small vanishing intervals

become.

(and for those who don't like "promo" ... too bad.

in my recent discussions with Paul it has now become an

essential term, and i'm quite certain the rest of you

will eventually see why.)

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> Dave, your last sentence is true. but -- and i'm assuming

> you're using "comma" in the general sense here -- you

> certainly can find a "comma" which exists between pitches

> at the two extremes of a meantone chain. those two pitches

> have irrational "ratios", and so does that "comma".

>

> for example, the interval between the origin and the

> 8ve-reduced 31st generator of 1/4-comma meantone, is

> 2^18 * 5^-(31/4) = ~ 6.068717548 cents. that "ratio"

> is definitely an irrational number ... and in fact,

> is the interval which vanishes in 31edo, and also

> which is *why* 31edo is such a good emulation of

> 1/4-comma meantone.

I would not suggest defining "promo" in a way which excludes this

example. If you take a monzo |A B C>, the promo is defined just by the

ratios A:B:C. You are talking about the same thing really whether A,

B, and C are restricted to being just integers, or are allowed to be

rational numbers. Traditionally, you let them be rational; it makes

the math a little easier. That would mean your "prime space" in which

a promo is said to be a "line" actually would be a (rational) vector

space, and the line actually would be a line. The 18:0:-31/4 ratios

and the 72:0:-31 ratios would then be simply two ways of denoting the

same promo.

> > Monz, if you could change those two ocurrences of

> > "pitch ratios" to "rational pitches" that should solve

> > the problem.

> well ... i don't think that's a good idea, as per my

> discussion above.

I suggest allowing them, at least, to denote promos.

> we need to discuss this more before i change that

> definition. perhaps it's a good idea to have separate

> terms to differentiate between rational "commas" and

> irrational ones.

Remember that we would be talking of a very special case of irrational

"commas" if we allowed fractional exponents. If we did the same

business with the Wilson fifth and 69-et, we would not end up with

something which could be discussed in terms of rational exponents, yet

2^40/Wilson^69, an interval of 1.5 cents, is certainly small enough to

be considered comatonse.

> > Me too. One suggestion I have is "komma".

> >

> > But whether using "komma" or "comma" to cover this

> > broader range of sizes that have the same range of

> > functions, it is well to spell out the fact that you

> > are doing so.

"Komma" is not a usable word in my book since it is simply an

alternative spelling, with the same sound.

> i propose "vapro", for "VAnishing PROmo".

I'd prefer just to say "promo", unless you forsee a use for projective

monzos which do not vanish.

on 8/13/04 10:56 PM, Dave Keenan <d.keenan@bigpond.net.au> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

>> I took a scan through the tuning dictionary and I really didn't

> find what I

>> was hoping to see, which is kind of a summary of all the different

> ways that

>> commas can be used.

> ...

>> Commas can be "left in" (not necessarily a piece of

> terminology), "tempered

>> out", "distributed" and perhaps many other possibilities.

>

> "Tempered out", "distributed", and "made to vanish" are all the same

> thing in regard to commas (and schismas, kleismas, dieses etc. if we

> disallow the generic use of "comma").

What do you mean by "same thing"? There are differences between these

things, right? Just checking. I think you mean something like as far as

the comma functions in defining the topology (?) of the resulting scale they

are all the same. But the topology (or whatever you want to call it) does

not determine the tuning. Right?

> By the way "anomaly" doesn't work for me as a substitute for this

> generic use of "comma" because of its implication of "a failure to

> meet expectations".

>

> If a comma (generic sense) is not tempered out (tempered to zero),

> then tempering may make it larger, smaller or even negative,

> relative to its untempered size, or it may remain untempered.

Making it larger seems an unlikely thing if you are going to collapse the

topology on that comma. The word tempering always implies a tuning, right?

Whereas a temperament as it is often used here is a class of tunings that

share a topology, right?

> If it is "left in" then it may be an actual scale step (not likely,

> but more likely for dieses and larger), or it may be a chromatic

> alteration of a scale notes when modulation occurs, or it may be

> symbolised as an accidental and used to notate scale notes even when

> the note without that accidental never occurs. Or it may not

> explicitly appear at all.

>

> There is a borderline case where although there is no tempering, a

> comma may already be so small that it can be ignored. So although it

> vanishes it isn't exactly _made_ to vanish. This is most likely for

> kleismas and schismas and smaller.

Yet if a piece makes use of this then the piece makes the comma vanish,

structurally speaking?

>> It seemed to me also that in the process of defining an actual

> tuning that

>> another concept of comma is perhaps introduced, but according to

> Carl this

>> isn't called a comma. When distributing a comma, which I think

> implies a

>> unison vector and also implies that the comma is rational (whereas

> the

>> historical use does not imply that, right?),

>

> Commas are always rational, ... until they are tempered. Hmm. That

> doesn't sound too clear. :-) A comma is always _defined_ as a

> rational frequency ratio, or equivalently as a prime-exponent-

> vector. You can't just pick any random number of cents and call it a

> comma.

Ok, I'll be interested to see how you reply to Monz's issue about this.

>> This

>> distributing reminds me of commas again, smaller commas, and my

> original

>> nieve intepretation of commas in temperament involved this perhaps

>> misfounded concept. I'm pretty vague on this, and welcome

> feedback.

>>

>

> You're right. It is misfounded. The quarter-comma of quarter-comma

> meantone, is not itself a comma, because it doesn't arise as the

> difference between two different stacks of rational intervals, and

> cannot be defined in rational terms. The fourth root of 81/80 is

> irrational.

Ah, is *that* what distributing usually refers to? I was thinking of

distributing the tempering among multiple dimensions of a lattice, but this

is distributing in a single dimension (though in anotheer sense multiple

dimensions are involved).

>> It seems to

>> me these comma functions apply regardless of the size of the

> comma, even if

>> they may seem a little absurd in some cases.

>

> Agreed. But I did include the deliberately vague

> proscription "typically smaller than a scale step".

>

>> So I'd almost wish there was

>> another word for this use of "comma",

>

> Me too. One suggestion I have is "komma".

If this functioning is primarily topological, isn't there already a

mathematical term for collapsing a topology according to a certain pattern,

and for the specification of the pattern over which the collapse is done?

Gene?

>> But we could also go on with this usage and when

>> necessary invoke some term like "generalized comma" to indicate

> that a comma

>> has gotten outside of its normal size range, and

> perhaps "functional comma"

>> to indicate that it is the function and not the size that is at

> stake.

>

> Those sound reasonable to me. The biggest disagreement is the one

> about what functions a comma may serve and still be a comma (or

> still be commatic). I haven't yet seen anything that would make me

> accept that "chromatic comma" (non-vanishing comma) is some kind of

> oxymoron.

>

>> Around here everyone will usually know this from context.

> Nonetheless for

>> interfacing with a larger world we may need some kind

> of "qualification"

>> terms which can be invoked when necessary, probably rarely.

>>

>> Has Paul dealt with this in his recent writings? I haven't had

> the time to

>> look at any of it yet.

>

> Paul Erlich has followed Paul Hahn in using "commatic" as the

> opposite of "chromatic". The _sound_ of these two words is certainly

> seductive in making one want them as opposites. "chromatic"

> versus "vanishing" certainly doesn't sound as nice.

Isn't "chromatic" also asking for trouble by still restricting the size?

Didn't Paul or someone come up with an example a couple months back of a

whole-tone sized comma?

Thanks for your inputs.

-Kurt

on 8/13/04 11:11 PM, Carl Lumma <ekin@lumma.org> wrote:

> Hi Kurt,

>

>> Maybe its too much to ask but

>> I'd love to see this all tied together in one comprehensive tutorial!

>

> Have you read Paul's papers?

I haven't had the time yet. Should I be quiet until I do? (I may be quite

for a long time. ;)

>> When distributing a comma, which I think implies a

>> unison vector

>

> As it happens, there's also very little agreement on what a

> "unison vector" is.

Yes, I think I need to start another thread about the vector issue.

> It's clear we've a long way to a consensus on this stuff, let alone

> a clear tutorial on it. My advice is to ignore terminology, and

> instead start working out tunings that interest you. In the process

> you will aquire the tools you need.

Yes, I think you're right. Like many others when first exposed to lingo

that is not well worked-out, I get confused trying to learn, and then I have

a little reaction to the chaos. But in the end I'm fine with ill-defined

terms because in the end all our words can appear insufficient for our

experiences, and so the art is in overcoming those apparent limits.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> on 8/13/04 11:11 PM, Carl Lumma <ekin@l...> wrote:

> > It's clear we've a long way to a consensus on this stuff, let

alone

> > a clear tutorial on it. My advice is to ignore terminology, and

> > instead start working out tunings that interest you. In the

process

> > you will aquire the tools you need.

>

> Yes, I think you're right. Like many others when first exposed to

lingo

> that is not well worked-out, I get confused trying to learn, and

then I have

> a little reaction to the chaos. But in the end I'm fine with ill-

defined

> terms because in the end all our words can appear insufficient for

our

> experiences, and so the art is in overcoming those apparent limits.

Carl and Kurt,

These are wise words indeed. Thank you both.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > Commas are always rational, ... until they are tempered.

> > Hmm. That doesn't sound too clear. :-) A comma is always

> > _defined_ as a rational frequency ratio, or equivalently

> > as a prime-exponent-vector. You can't just pick any random

> > number of cents and call it a comma.

>

>

>

> Dave, your last sentence is true. but -- and i'm assuming

> you're using "comma" in the general sense here -- you

> certainly can find a "comma" which exists between pitches

> at the two extremes of a meantone chain. those two pitches

> have irrational "ratios", and so does that "comma".

>

> for example, the interval between the origin and the

> 8ve-reduced 31st generator of 1/4-comma meantone, is

> 2^18 * 5^-(31/4) = ~ 6.068717548 cents. that "ratio"

> is definitely an irrational number ... and in fact,

> is the interval which vanishes in 31edo, and also

> which is *why* 31edo is such a good emulation of

> 1/4-comma meantone.

>

>

>

> > > The tonalsoft comma page currently relies heavily on

> > > a quote from Dave Keenan which gives two definitions of

> > > comma. However neither definition seems to say much

> > > about the functions of commas as commonly discussed on

> > > this list,

> >

> > No, nothing about functions, sorry.

> >

> > > nor is any restriction to rational values mentioned.

> >

> > Technically, you're right! Yikes! But it was certainly my

> > intention to so restrict it.

> >

> > Monz, if you could change those two ocurrences of

> > "pitch ratios" to "rational pitches" that should solve

> > the problem.

>

>

> well ... i don't think that's a good idea, as per my

> discussion above.

But the comma you refer to is tempered. That's the only reason it's

irrational. Untempered, it may be expressed as 3^31 / 2^49 or any

number of other rationals. So this is not a counterexample.

> a totally new word would be better, and i will

> continue to maintain that position in the face

> of all contrary arguments.

That sounds sadly like dogmatism or fanaticism.

> which strengthens my argument that we really need a

> term to designate the "vanishingness" of a small interval.

>

> i propose "vapro", for "VAnishing PROmo".

Ah Monz, ... you're a lunatic ... but a lovable one. :-)

Newbie: What's a "vapro"?

Old hand: It's a "vanishing promo"?

Newbie: What's a "promo"?

Old hand: Well it's either something you do to promote yourself, or

it's a "projective monzo".

Newbie: What's a "monzo"?

Old hand: Well it's either a prime exponent vector, or it's one of

the guys that made up this jargon.

Newbie: Oh. Now I get why it's a promo.

;-)

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > "Tempered out", "distributed", and "made to vanish" are all the

same

> > thing in regard to commas (and schismas, kleismas, dieses etc.

if we

> > disallow the generic use of "comma").

>

> What do you mean by "same thing"? There are differences between

these

> things, right?

Not that I can think of. Commas never really vanish, they either get

spread so thinly (by being distributed over multiple intervals) so

that you don't notice them, or you _decide_ not to notice them.

That's what tempering is.

> Just checking. I think you mean something like as far as

> the comma functions in defining the topology (?) of the resulting

scale they

> are all the same. But the topology (or whatever you want to call

it) does

> not determine the tuning. Right?

You've lost me here. Sorry.

> Making it larger seems an unlikely thing if you are going to

collapse the

> topology on that comma.

Sure. But who said we were? When we make some commas vanish, others

will get larger.

> The word tempering always implies a tuning, right?

Not sure I understand the question? Tempering is the process of

ditributing commas and the result is a temperament.

> Whereas a temperament as it is often used here is a class of

tunings that

> share a topology, right?

I guess so.

> Yet if a piece makes use of this then the piece makes the comma

vanish,

> structurally speaking?

I guess so. Like I said, it's borderline.

> Ah, is *that* what distributing usually refers to? I was thinking

of

> distributing the tempering among multiple dimensions of a lattice,

but this

> is distributing in a single dimension (though in anotheer sense

multiple

> dimensions are involved).

You're quite right in both cases. This might help:

http://dkeenan.com/Music/DistributingCommas.htm

> Isn't "chromatic" also asking for trouble by still restricting the

size?

I don't see it as restricting the size. But you may be right.

We're generalising from "chromatic" meaning "not in the diatonic

scale" or "indicated by a sharp or flat accidental outside of the

key signature" to anything that is not in the "usual" scale of a

given temperament and is indicated by _any_ accidental, not just a

sharp or flat.

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> > > > The tonalsoft comma page currently relies heavily on

> > > > a quote from Dave Keenan which gives two definitions of

> > > > comma. However neither definition seems to say much

> > > > about the functions of commas as commonly discussed on

> > > > this list,

> > >

> > > No, nothing about functions, sorry.

> > >

> > > > nor is any restriction to rational values mentioned.

> > >

> > > Technically, you're right! Yikes! But it was certainly my

> > > intention to so restrict it.

> > >

> > > Monz, if you could change those two ocurrences of

> > > "pitch ratios" to "rational pitches" that should solve

> > > the problem.

> >

> >

> > well ... i don't think that's a good idea, as per my

> > discussion above.

>

> But the comma you refer to is tempered. That's the only reason it's

> irrational. Untempered, it may be expressed as 3^31 / 2^49 or any

> number of other rationals. So this is not a counterexample.

hmm ... ok, well ... in any case, it's far to late (early?)

for me to think about this stuff anymore "tonight".

my brain is fried.

i think we need some more discussion on this one.

> > a totally new word would be better, and i will

> > continue to maintain that position in the face

> > of all contrary arguments.

>

> That sounds sadly like dogmatism or fanaticism.

yeah, upon reading it again it sounds like that

to me too. thanks for calling me on that.

i didn't really have to make such a strong stand

anyway ... all i had to do was make up a word i like

and put it into the Encyclopaedia. ;-PP

... which is exactly what i've done anyway.

> > which strengthens my argument that we really need a

> > term to designate the "vanishingness" of a small interval.

> >

> > i propose "vapro", for "VAnishing PROmo".

>

> Ah Monz, ... you're a lunatic ... but a lovable one. :-)

kisses all around from me too!

> Newbie: What's a "vapro"?

>

> Old hand: It's a "vanishing promo"?

>

> Newbie: What's a "promo"?

>

> Old hand: Well it's either something you do to promote yourself, or

> it's a "projective monzo".

>

> Newbie: What's a "monzo"?

>

> Old hand: Well it's either a prime exponent vector, or it's one of

> the guys that made up this jargon.

>

> Newbie: Oh. Now I get why it's a promo.

>

> ;-)

ok, wiseguy ... how about:

Newbie: What's a "vapro"?

Old hand:

http://tonalsoft.com/enc/index2.htm?vapro.htm

Newbie: What's a "promo"?

Old hand:

http://tonalsoft.com/enc/index2.htm?promo.htm

Newbie: What's a "monzo"?

Old hand:

http://tonalsoft.com/enc/index2.htm?monzo.htm

if i had somebody around who could have given me

answers like that 10 years ago, i would have saved

myself a ridiculous amount of time and effort.

-monz

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

>

> > > "Tempered out", "distributed", and "made to vanish"

> > > are all the same thing in regard to commas (and schismas,

> > > kleismas, dieses etc. if we disallow the generic use of

> > > "comma").

> >

> > What do you mean by "same thing"? There are differences

> > between these things, right?

>

> Not that I can think of. Commas never really vanish, they

> either get spread so thinly (by being distributed over

> multiple intervals) so that you don't notice them, or you

> _decide_ not to notice them. That's what tempering is.

i can't bring myself to say that they don't really vanish.

if one makes a lattice in a certain dimensionality of

prime-space, then distributes the promo ("comma") so

that it cannot be noticed (thus making it a vapro),

then warps the lattice to show the fact that it vanishes,

it really does physically vanish from the lattice!

that's exactly how Musica made these 4-strand helical

meantone lattices:

http://tonalsoft.com/enc/index2.htm?meantone.htm&helix

(wait for it to load all the way and it will take you

right to those lattices.)

> > Whereas a temperament as it is often used here is a

> > class of tunings that share a topology, right?

>

> I guess so.

i think that might be a good description of a temperament

"family". but i'll defer here to others who know more

about topology.

-monz

>> "Tempered out", "distributed", and "made to vanish" are all the same

>> thing in regard to commas (and schismas, kleismas, dieses etc. if we

>> disallow the generic use of "comma").

>

>What do you mean by "same thing"? There are differences between these

>things, right?

No. These phrases are usually used interchangeably.

>> There is a borderline case where although there is no tempering, a

>> comma may already be so small that it can be ignored. So although it

>> vanishes it isn't exactly _made_ to vanish. This is most likely for

>> kleismas and schismas and smaller.

>

>Yet if a piece makes use of this then the piece makes the comma vanish,

>structurally speaking?

Yes.

>> Commas are always rational, ... until they are tempered. Hmm. That

>> doesn't sound too clear. :-) A comma is always _defined_ as a

>> rational frequency ratio, or equivalently as a prime-exponent-

>> vector. You can't just pick any random number of cents and call it a

>> comma.

That's what Dave's saying. I'm not sure it's strictly true in

the literature, though.

>> You're right. It is misfounded. The quarter-comma of quarter-comma

>> meantone, is not itself a comma, because it doesn't arise as the

>> difference between two different stacks of rational intervals, and

>> cannot be defined in rational terms. The fourth root of 81/80 is

>> irrational.

>

>Ah, is *that* what distributing usually refers to? I was thinking of

>distributing the tempering among multiple dimensions of a lattice, but

>this is distributing in a single dimension (though in anotheer sense

>multiple dimensions are involved).

Distributing is distributing, any way you'd like to do it!

>>> But we could also go on with this usage and when necessary

>>> invoke some term like "generalized comma" to indicate

>>> that a comma has gotten outside of its normal size range,

>>> and perhaps "functional comma" to indicate that it is the

>>> function and not the size that is at stake.

>>

>> Those sound reasonable to me. The biggest disagreement is the one

>> about what functions a comma may serve and still be a comma (or

>> still be commatic). I haven't yet seen anything that would make me

>> accept that "chromatic comma" (non-vanishing comma) is some kind of

>> oxymoron.

"Chromatic comma" is fine. "Comma" does not usually imply vanishing,

though Gene wanted to make it so.

>Isn't "chromatic" also asking for trouble by still restricting the

>size? Didn't Paul or someone come up with an example a couple months

>back of a whole-tone sized comma?

Dave is one of a vast minority of theorists who worries about

defining rigid size constraints for things like commas.

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> If this functioning is primarily topological, isn't there already a

> mathematical term for collapsing a topology according to a certain

pattern,

> and for the specification of the pattern over which the collapse is

done?

> Gene?

Don't ask me. Words such as "topology" and "topologicial" have

specific mathematical meanings, and the way you are using them is

making me dizzy. Can you explain what you mean more precisely, and

without using "topology" or "topological"?

> Isn't "chromatic" also asking for trouble by still restricting the size?

> Didn't Paul or someone come up with an example a couple months back of a

> whole-tone sized comma?

I've just employed 125/108 as a comma for a Fokker block with great

popular sucess; 125/108 is a half-fourth sized sub-subminor third of

253 cents.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> i can't bring myself to say that they don't really vanish.

But in any case, you agree that making them vanish, distributing

them, and tempering them out are all the same thing?

> if one makes a lattice in a certain dimensionality of

> prime-space, then distributes the promo ("comma") so

> that it cannot be noticed (thus making it a vapro),

> then warps the lattice to show the fact that it vanishes,

> it really does physically vanish from the lattice!

Yes. It vanishes from the lattice, but the lattice isn't physical,

it's a mathematical abstraction. What's physical is what you hear,

or measure in Hertz or cents.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> "Chromatic comma" is fine. "Comma" does not usually imply

vanishing,

> though Gene wanted to make it so.

Actually I think Gene got that from Paul Erlich, who got it from

Paul Hahn via the term "commatic" being juxtaposed

against "chromatic".

Also, Gene says his "kernel elements of a temperament" got

translated to "commas of a temperament", and in that context,

there's not too much potential for confusion since a temperament's

vanishing commas are more special than its chromatic ones. The

former can define the temperament whereas the latter are somewhat

arbitrary, depending as they do on a choice of a "white-note" scale.

> >Isn't "chromatic" also asking for trouble by still restricting the

> >size? Didn't Paul or someone come up with an example a couple

months

> >back of a whole-tone sized comma?

>

> Dave is one of a vast minority of theorists who worries about

> defining rigid size constraints for things like commas.

Yes. But that isn't relevant here. Kurt seems to be assuming

that "chromatic comma" means a comma around a semitone in size (so

generic comma, not the approx 20 cents range), but the intended

meaning of chromatic here is "functioning as an alteration in pitch

from a "standard" scale of some temperament, or even from

Pythagorean".

>> >Isn't "chromatic" also asking for trouble by still restricting the

>> >size? Didn't Paul or someone come up with an example a couple

>> >months back of a whole-tone sized comma?

>>

>> Dave is one of a vast minority of theorists who worries about

>> defining rigid size constraints for things like commas.

>

>Yes. But that isn't relevant here. Kurt seems to be assuming

>that "chromatic comma" means a comma around a semitone in size (so

>generic comma, not the approx 20 cents range), but the intended

>meaning of chromatic here is "functioning as an alteration in pitch

>from a "standard" scale of some temperament, or even from

>Pythagorean".

Yes, sorry about that. I misconstrued things.

-Carl

Gene,

on 8/14/04 11:47 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

>

>> If this functioning is primarily topological, isn't there already a

>> mathematical term for collapsing a topology according to a certain

> pattern,

>> and for the specification of the pattern over which the collapse is

> done?

>> Gene?

>

> Don't ask me. Words such as "topology" and "topologicial" have

> specific mathematical meanings, and the way you are using them is

> making me dizzy. Can you explain what you mean more precisely, and

> without using "topology" or "topological"?

Sure I can avoid this easily. I think we can approach what I was getting at

by defining "temperament" more clearly. The tonalsoft dictionary doesn't

seem to define it the way I expect it to be defined for usage in this group,

as I understand it, partly from talking to Carl about it recently.

Maybe I can start with a question. What uniquely determines a temperament?

Once I have this answer then I can continue with more clarity.

Well I can give more hints than that. This is about the distinction between

temperament and tuning. Traditionally a temperament was a tuning. On this

list, in relation to "comma theory" this is not the case. I think a

temperament is uniquely defined by the commas tempered out. But this does

not uniquely define a tuning, right? Once this is confirmed, I can go on.

-Kurt

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> if i had somebody around who could have given me

> answers like that 10 years ago, i would have saved

> myself a ridiculous amount of time and effort.

Yes Monz,

As I've said many times, I don't know what we'd do without your

encyclopedia. But don't you think that when an encyclopedia that's

edited by one person starts having entries for new terms added at

the whim of that editor and deleted again within a few days and new

terms put in their place, it seriously undermines the authority of

that encyclopedia. Why should anyone take any notice of any of it if

that sort of thing can happen?

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Maybe I can start with a question. What uniquely determines a

temperament?

I would say a regular temperament is uniquely determined by a

homomorphic mapping from the p-limit, or possibly another

finitely-generated subgroup of the positive rationals, to an abstract

free group of smaller rank. This can be specified by giving a wedgie,

a kernel, or an explicit mapping.

Note I do not include the tuning map as part of the definition, so

this is an abstract definition of what a regular temperament is.

However, this is how the word is most commonly used; people may object

to it but the same people talk of 1/4-comma meantone or 2/7-comma

meantone as if they were both meantone. My definition also says that

even though 31-et meantone is tuned to a group of rank one, it is

still qua meantone a group of rank two, and the tuning mapping is

another issue.

You might also note that this definition, which says a temperament is

a morphism, makes no sense unless you get the category right, which

connects to the thread about spaces.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> I would say a regular temperament is uniquely determined by a

> homomorphic mapping from the p-limit, or possibly another

> finitely-generated subgroup of the positive rationals, to an abstract

> free group of smaller rank. This can be specified by giving a wedgie,

> a kernel, or an explicit mapping.

I should have said it is determined by the mapping, and uniquely

determined by the wedgie or kernel.

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > i can't bring myself to say that they don't really vanish.

>

> But in any case, you agree that making them vanish, distributing

> them, and tempering them out are all the same thing?

>

> > if one makes a lattice in a certain dimensionality of

> > prime-space, then distributes the promo ("comma") so

> > that it cannot be noticed (thus making it a vapro),

> > then warps the lattice to show the fact that it vanishes,

> > it really does physically vanish from the lattice!

>

> Yes. It vanishes from the lattice, but the lattice isn't physical,

> it's a mathematical abstraction. What's physical is what you hear,

> or measure in Hertz or cents.

and that is exactly *my* point. apparently we're

agreeing with each other from opposite sides of a tall

fence, which is blocking the view into each other's reasoning.

:)

for example,

you can listen to music in 12edo all day long and you

will never *hear* a syntonic-comma, or a diesis, or

a skhisma, or a pythagorean-comma.

yes, i composer might be able to figure out how to

play around with his listener's expectations so that

the listeners *think* they are going to hear one of

those small intervals ... but if the tuning is 12edo,

those intervals will never be heard because they

do not physically exist in that tuning.

in 5-limit JI, with an 8ve-equivalent lattice, you can

play the ratio which plots to 5^1 on the lattice

(ratio 5:4, monzo [-2 0, 1>) and you can play the

ratio which plots to 3^4 (ratio 81:64, monzo [-6 4, 0>),

and if you're listening carefully you will hear the

difference of a syntonic comma.

but in 12edo, both of these JI "targets" are mapped

to the same 12edo note. so no matter whether you

consider 2^(4/12) to map 3^4 or to map 5^1, listen

as carefully as you wish and you will never hear any

difference. the syntonic comma is not there,

and won't be found between any two pitches in 12edo.

-monz

>I would say a regular temperament is uniquely determined by a

>homomorphic mapping from the p-limit, or possibly another

>finitely-generated subgroup of the positive rationals, to an abstract

>free group of smaller rank. This can be specified by giving a wedgie,

>a kernel, or an explicit mapping.

//

>You might also note that this definition, which says a temperament is

>a morphism, makes no sense unless you get the category right, which

>connects to the thread about spaces.

Yes, you may indeed have a point here.

-Carl

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> > Yes. It vanishes from the lattice, but the lattice isn't

physical,

> > it's a mathematical abstraction. What's physical is what you

hear,

> > or measure in Hertz or cents.

>

>

> and that is exactly *my* point. apparently we're

> agreeing with each other from opposite sides of a tall

> fence, which is blocking the view into each other's reasoning.

>

> :)

>

>

> for example,

>

> you can listen to music in 12edo all day long and you

> will never *hear* a syntonic-comma, or a diesis, or

> a skhisma, or a pythagorean-comma.

You can't hear a whole syntonic comma, because it's been cut into

pieces and spread around the place. But you can certainly hear these

pieces, not as melodic steps of course, but as roughness or beating

in the harmonies. So they have not actually _vanished_ from what you

hear.

...

> difference. the syntonic comma is not there,

> and won't be found between any two pitches in 12edo.

This is hardly the point. One can have a completely untempered JI

scale, e.g. Pythagorean 7 or 12, in which the syntonic comma can't

be found between any two pitches. We don't say that the syntonic

comma has been tempered out or made to vanish in this scale.

hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>

wrote:

> > > Yes. It vanishes from the lattice, but the lattice

> > > isn't physical, it's a mathematical abstraction. What's

> > > physical is what you hear, or measure in Hertz or cents.

> >

> >

> > and that is exactly *my* point. apparently we're

> > agreeing with each other from opposite sides of a tall

> > fence, which is blocking the view into each other's reasoning.

> >

> > :)

> >

> >

> > for example,

> >

> > you can listen to music in 12edo all day long and you

> > will never *hear* a syntonic-comma, or a diesis, or

> > a skhisma, or a pythagorean-comma.

>

> You can't hear a whole syntonic comma, because it's been cut into

> pieces and spread around the place. But you can certainly hear these

> pieces, not as melodic steps of course, but as roughness or beating

> in the harmonies. So they have not actually _vanished_ from what you

> hear.

>

> ...

> > difference. the syntonic comma is not there,

> > and won't be found between any two pitches in 12edo.

>

> This is hardly the point. One can have a completely untempered JI

> scale, e.g. Pythagorean 7 or 12, in which the syntonic comma can't

> be found between any two pitches. We don't say that the syntonic

> comma has been tempered out or made to vanish in this scale.

but the syntonic-comma doesn't have anything to do

with a pythagorean tuning! you'll never find it

there either.

anyway, the thing you wrote about roughness in 12edo makes

your point clear. now i'm understanding you.

-monz

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

>

> > Maybe I can start with a question. What uniquely determines a

> temperament?

This is an excellent question.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

> I would say a regular temperament is uniquely determined by a

> homomorphic mapping from the p-limit, or possibly another

> finitely-generated subgroup of the positive rationals, to an

abstract

> free group of smaller rank. This can be specified by giving a

wedgie,

> a kernel, or an explicit mapping.

To many people on this list, like me, who haven't recently used or

studied abstract algebra, category theory, geometric algebra, and

Gene-Ward-Smith-ese (for "wedgie"), the above is probably about as

meaningful as this:

"A regular temperament is uniquely determined by a

heteromagic mapping from the p-limit, or possibly another

finitely-gargled sub-band of the positive rationals, to an abstract

free band of lesser authority. This can be specified by pulling

someone's underpants real tight, a peanut, or an explicit mapping."

What if we started with the idea that a temperament was something

physical, that you can hear, or at least measure as a set of

frequencies.

The Shorter Oxford has this entry for the musical meaning

of "temperament":

"The adjustment of intervals of the scale (in the tuning of

instruments of fixed intonation, as keyboard instruments), so as to

adapt them to purposes of practical harmony : consisting in slight

variations of the pitch of the notes from true or 'just' intonation,

in order to make them available in different keys; a particular

system for doing this. (Sometimes extended to any system of tuning.)

1727."

The example text they quote is:

"The chief temperaments ... are mean-tone temperament and equal

temperament (now almost universal), in which the octave is divided

into twelve (theoretically) equal semitones, so that the variations

of pitch are evenly distributed throughout all keys."

We have generalised this idea beyond 5-limit and beyond 7-tone or 12-

tone chain-of-fifth scales in a fairly obvious way, about which

there is no dispute that I am aware of.

But something we haven't agreed on is the following recasting of

Kurt's question: "How can we tell, by listening or measuring,

whether two tunings are different temperaments, or are merely

slightly different tunings of the same temperament?"

To the mathematicians, the mapping from generators to prime number

intervals is everything. But you can't always obtain a unique

mapping by reverse-engineering the tuning. This was a very real

problem with claims made about the Zeng Bells a while back. More

than one mapping can result in the same tuning. Do we really want to

call them different temperaments if you can't hear or measure any

difference in the product? Maybe we do.

But notice that we don't do this with one dimensional (equal)

temperaments. Even something as simple as 24-ET has more than one

meaningful mapping for the primes number 7. But we don't refer to

these two mappings as two different temperaments. There is only one

24 tone equal temperament.

I will understand if you want to apply to me Bishop Berkeley's

observation on philosphers, "They first raise a dust, and then

complain that they cannot see." :-)

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> What if we started with the idea that a temperament was something

> physical, that you can hear, or at least measure as a set of

> frequencies.

And when the Shorter Oxford Dictionary speaks of "meantone", what do

they mean? How some given piano is tuned on a given day--the Standard

Meantone Piano, held in a climate controlled room in Paris? When it

says the interval differences of equal temperament are "theoretically"

equal, is it not denying that some specific physical piano is being

talked about at all? "Theoretically equal" is an idealization. It

isn't something physical that you can hear, or measure as a set of

frequencies, it is a theoretical construct.

Using your system, you cannot name or classify temperaments. You can

have no theory, because you can have no theoretical constructs. You

can measure what a violinist does when he or she plays a score, but

the score itself is too abstract for your point of view to deal with;

the violinists who first played it may well be long dead, and no

recording of 18th century violin playing will allow you to measure

what was done in the 18th century.

> But notice that we don't do this with one dimensional (equal)

> temperaments. Even something as simple as 24-ET has more than one

> meaningful mapping for the primes number 7. But we don't refer to

> these two mappings as two different temperaments.

Wake up and smell the coffee. I certainly do, and I don't think I am

the only one.

There is only one

> 24 tone equal temperament.

There is *no* "24 tone equal temperament" by your logic. There are

only physically existing systems, producing sound by one means or

another, which may reasonably be considered instances of more or less

equal division of the octave into 24 parts. It is simply a way of

evaluating tunings. You can't ask what temperaments 24-equal supports,

because the question is meaningless. You can't write a score in it, as

to do that assumes quarter-tones are not necessarily physically

related to some particular place, time, and object. You certainly will

have grave difficulty coming up with any theory of temperaments;

including the sorts of theories you've put forward. Those are straight

out the door and in the trash now.

>> I would say a regular temperament is uniquely determined by a

>> homomorphic mapping from the p-limit, or possibly another

>> finitely-generated subgroup of the positive rationals, to an

>> abstract free group of smaller rank. This can be specified by

>> giving a wedgie, a kernel, or an explicit mapping.

>

>To many people on this list, like me, who haven't recently used or

>studied abstract algebra, category theory, geometric algebra, and

>Gene-Ward-Smith-ese (for "wedgie"), the above is probably about as

>meaningful as this:

>

>"A regular temperament is uniquely determined by a

>heteromagic mapping from the p-limit, or possibly another

>finitely-gargled sub-band of the positive rationals, to an abstract

>free band of lesser authority. This can be specified by pulling

>someone's underpants real tight, a peanut, or an explicit mapping."

ROTFL! :)

Though it meant a little more than this to me. Let me step

through Gene's definition...

regular temperament - The thing we're defining, which is

the target of a...

homomorphic mapping - I'd looked this up before, but I had to

do it again. It looks like this phrase could be replaced with

"homomorphism", since a morphism is already a type of mapping.

And homomorphism is just a "general morphism". It isn't clear

what kinds of mappings qualify as general. I'm assuming the

target of the map is expected to be the same sort of thing you

started with. For example, set -> set, group -> group, etc.

But it looks like we're dealing with groups here, so I think

Gene may have meant "group homomorphism", which means not only

do you wind up with a group, but that group has the same identity

element and the same operator as the group you started with.

...from the...

p-limit - Prime limit. Most list members are familiar with

these.

...or...

finitely-generated subgroup of the positive rationals - I

assume this allows for odd-limit and/or non-consective prime

bases like {2, 3, 7 11}.

...to an...

abstract free group of smaller rank - A group is a set that's

closed under some operator, has an identity element, and a law

of inverses. This was actually discussed in my 8th-grade

Algebra class. I'm not sure about the "free" part. It sounds

like it's meant to rule out torsion. The "abstract" part

looks superfluous to me. I think rank is just what we might

call dimension -- the number of generators needed, or more

precisely, the minimum number of elements needed to cover the

group using our operator. So by saying "smaller rank" we just

mean we expect temperament to simplify things.

...and can be named by any of...

wedgie - Gene and Graham have explained these. IIRC they

suck numbers out of the map and put them into a single

n-tuple in a certain way.

kernel - I haven't a clue.

explicit mapping - I think most tuning-math denizens are

familiar with these, though I've only worked with those

beloning to linear temperaments.

>But notice that we don't do this with one dimensional (equal)

>temperaments. Even something as simple as 24-ET has more than one

>meaningful mapping for the primes number 7. But we don't refer to

>these two mappings as two different temperaments. There is only one

>24 tone equal temperament.

Gene blew apart my thinking on this years ago, with this very

example, when I asked him why he doesn't care about consistency.

The answer is that all regular temperaments are consistent, and

that ETs are not necessarily temperaments. I haven't been able

to enjoy domestic beer since. :)

-Carl

Dave Keenan wrote:

> To the mathematicians, the mapping from generators to prime number > intervals is everything. But you can't always obtain a unique > mapping by reverse-engineering the tuning. This was a very real > problem with claims made about the Zeng Bells a while back. More > than one mapping can result in the same tuning. Do we really want to > call them different temperaments if you can't hear or measure any > difference in the product? Maybe we do.

> > But notice that we don't do this with one dimensional (equal) > temperaments. Even something as simple as 24-ET has more than one > meaningful mapping for the primes number 7. But we don't refer to > these two mappings as two different temperaments. There is only one > 24 tone equal temperament.

This is an interesting point: the difference between "bug" and "superpelog" for instance is mainly in the mapping, and in the number of generators in a typical scale. "Superpelog" is basically a 14-note scale with the mapping [<1 2 1 3|, <0 -2 6 -1|], while the "bug" mapping [<1 2 3 3|, <0 -2 -3 -1| is only good up to 11 notes, and a 9-note scale is typical. But the generator / period ratio is very similar: 260.76 / 1206.55 for 7-limit superpelog, 260.26 / 1200.00 for 5-limit bug, and 254.90 / 1194.64 for 7-limit bug. So audibly they're pretty similar tunings; the difference is in which notes are used to approximate consonant intervals. This is similar to the use of two different approximate fifths in ET's like 23-ET or 64-ET.

Another way to categorize tunings is by putting them on a branch of the scale tree (by their generator / period ratio if the period is around an octave). This would lump together scales that have different mappings, but sound roughly the same. For each specific generator / period ratio, a number of different mappings can be assigned depending on how far you continue the chain of generators in one direction or the other. One thing this could be useful for is notation: it would make sense to use the same notation for all scales with roughly the same g/p ratio (like meantone 504.13 / 1201.7 and flattone 507.14 / 1202.54, or dominant 495.88 / 1195.23 and garibaldi 498.12 / 1200.76).

meantone: 12&19, [<1, 2, 4, 7|, <0, -1, -4, -10|]

flattone: 19&26, [<1, 2, 4, -1|, <0, -1, -4, 9|]

dominant: 5&12, [<1, 2, 4, 2|, <0, -1, -4, 2|]

garibaldi: 12&29, [<1, 2, -1, -3|, <0, -1, 8, 14|]

>> To the mathematicians, the mapping from generators to prime number

>> intervals is everything. But you can't always obtain a unique

>> mapping by reverse-engineering the tuning. This was a very real

>> problem with claims made about the Zeng Bells a while back. More

>> than one mapping can result in the same tuning. Do we really want to

>> call them different temperaments if you can't hear or measure any

>> difference in the product? Maybe we do.

//

>This is an interesting point: the difference between "bug" and

>"superpelog" for instance is mainly in the mapping, and in the number of

>generators in a typical scale. "Superpelog" is basically a 14-note scale

>with the mapping [<1 2 1 3|, <0 -2 6 -1|], while the "bug" mapping [<1 2

>3 3|, <0 -2 -3 -1| is only good up to 11 notes, and a 9-note scale is

>typical. But the generator / period ratio is very similar: 260.76 /

>1206.55 for 7-limit superpelog, 260.26 / 1200.00 for 5-limit bug, and

>254.90 / 1194.64 for 7-limit bug. So audibly they're pretty similar

>tunings; the difference is in which notes are used to approximate

>consonant intervals. This is similar to the use of two different

>approximate fifths in ET's like 23-ET or 64-ET.

Erv Wilson has even suggested that the meantone and schismic

mappings may be separate in 12-tET, to the point of being able

to hear the difference.

Gene may have something to say about using tunings to categorize

temperaments -- he tried it initially but lately has been headed

toward using comma sequences. Gene, can you refresh us on the

gotchas of the tunings-based approach?

-Carl

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> Another way to categorize tunings is by putting them on a branch of the

> scale tree (by their generator / period ratio if the period is

around an

> octave).

To actually use this to categorize you'd have to use a standard

version of the generator, such as the 1 < g < sqrt(period).

This would lump together scales that have different mappings,

> but sound roughly the same. For each specific generator / period ratio,

> a number of different mappings can be assigned depending on how far you

> continue the chain of generators in one direction or the other.

And also depending on what sort of numbers--for example, which prime

limit--we had in mind.

One

> thing this could be useful for is notation: it would make sense to use

> the same notation for all scales with roughly the same g/p ratio (like

> meantone 504.13 / 1201.7 and flattone 507.14 / 1202.54, or dominant

> 495.88 / 1195.23 and garibaldi 498.12 / 1200.76).

Where do you switch? Why, for instance, don't you continue on to

superpyth?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Gene may have something to say about using tunings to categorize

> temperaments -- he tried it initially but lately has been headed

> toward using comma sequences. Gene, can you refresh us on the

> gotchas of the tunings-based approach?

I was suggesting both of these in connection to family relationships.

Which 7-limit temperament, if any, should we simply call "meantone"?

Is there an 11-limit temperament we might simply call "meantone" also?

My suggestion was to sort it out, to the extent possible, excluding

anything too awful and then using TOP tuning.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> homomorphic mapping - I'd looked this up before, but I had to

> do it again. It looks like this phrase could be replaced with

> "homomorphism", since a morphism is already a type of mapping.

> And homomorphism is just a "general morphism".

Actually it's more like a morphism is a general homomorphism. Category

was (and occasionally still is) called "generalized abstract nonsense

theory" for a reason.

> finitely-generated subgroup of the positive rationals - I

> assume this allows for odd-limit and/or non-consective prime

> bases like {2, 3, 7 11}.

It doesn't distinguish the 7-limit from the 9-limit, but does allow

{2,3,7,11} and many other possibilities, including many hardly ever

even considered, much less adopted.

> ...to an...

>

> abstract free group of smaller rank - A group is a set that's

> closed under some operator, has an identity element, and a law

> of inverses. This was actually discussed in my 8th-grade

> Algebra class.

Ha! I thought they killed the New Math.

I'm not sure about the "free" part. It sounds

> like it's meant to rule out torsion.

It rules out torsion, it also says you can't keep subdividing, so it

rules out vector spaces.

http://en.wikipedia.org/wiki/Free_abelian_group

The "abstract" part

> looks superfluous to me.

I just wanted to make clear that tuning was not a part of the

definition at this point.

> kernel - I haven't a clue.

The kernel of a temperament is all of the intervals it sends to the

unison; these form another group.

>> abstract free group of smaller rank - A group is a set that's

>> closed under some operator, has an identity element, and a law

>> of inverses. This was actually discussed in my 8th-grade

>> Algebra class.

>

>Ha! I thought they killed the New Math.

My hillbilly school district couldn't afford to update their

textbooks. :)

This was actually the first chapter in the book, which I remember

as my favorite part of the class, but which all the other students

seemed to hate.

>> I'm not sure about the "free" part. It sounds

>> like it's meant to rule out torsion.

>

>It rules out torsion, it also says you can't keep subdividing, so it

>rules out vector spaces.

>

> http://en.wikipedia.org/wiki/Free_abelian_group

It looks like this "free" business is behind the "fundamental

theorem of arithmetic".

>> kernel - I haven't a clue.

>

>The kernel of a temperament is all of the intervals it sends to the

>unison; these form another group.

Aha! This explains why I've never seen a kernel written out.

For some reason I missed it here:

http://mathworld.wolfram.com/GroupKernel.html

-Carl

Hi Carl,

>> I would say a regular temperament is uniquely determined by a

>> homomorphic mapping from the p-limit, or possibly another

>> finitely-generated subgroup of the positive rationals, to an

>> abstract free group of smaller rank. This can be specified by

>> giving a wedgie, a kernel, or an explicit mapping.

>

>To many people on this list, like me, who haven't recently used or

>studied abstract algebra, category theory, geometric algebra, and

>Gene-Ward-Smith-ese (for "wedgie"), the above is probably about as

>meaningful as this:

>

This really isn't so mysterious at all if one is accustomed

to the mathematical terms that Gene is using here.

An abstract free group of smaller rank just

is an algebraic way of talking about

a lattice with fewer lattice directions.

That's by a standard result that a

finite free algebraic group is isomorphic to

Z^n (in algebraic lanaguage).

So wherever you see abstract free group

you can substitute the word lattice,

with no loss of understanding

(though an algebraicist might complain that

ones understanding of what is happening

isn't sufficiently abstract :-) ).

Its rank just means the dimension of

the lattice.

So a more geometrical way of saying the same

thing is:

A regular temperament is uniquely determined

by an interval addition preserving mapping

of a p-limit lattice to one of lower

dimension.

By interval addition preserving, this means that

e.g. if 5/4 * 6/5 = 3/2 and you temper it to

the pythagorean scale, you want to map it

to intervals such as 81/64, 32/27 and 3/2

with (81/64) * (32/27) = 3/2

Then for the rest of what it says:

You can either give the mapping explicitly,

or you can say which of the elements

map to the unison vector (this is what

is meant by the "kernel")

- which can then be extended to give a mapping

of the entire lattice

- or you can use a wedge product

- now that last bit I don't understand

quite yet, how he does these mappings

using wedge products, I seem to be missing

something. But the rest is clear.

I don't understand why he hasn't mentioned

what seems the most obvious way to do

it - that you just need to give a mapping

of the bases of the lattice

- if you say where 2, 3, and 5 map to

then you can say where every interval

maps to.

So for instance if 5/4 goes to 81/64

then 5/1 goes to 81/16

3/1 then just goes to 3/1

and if you require addition of intervals

to be preserved, then that

specifies an entire mapping

of the five limit lattice to the three

limit one - there is no room for further

choice having set out those requirements

on the mapping. A map that is defined

over the entire lattice must be defined

on its basis vectores 2, 3, 5 etc

so this procedure can always be followed

- unless I'm missing something.

At any rate I'm pretty sure that is what

it is saying - correct me if I'm wrong Gene!

And I'd be interested in a newbie explanation

of how wedge products are used to make

these mappings and to study them as that

seems to be one thing I'm missing right

now in my understanding of the

encyclopedia. It is probably some simple thing

like not understanding that vals

were being used to count the number of scale

degrees spanned by an interval.

Robert

Hi Robert,

>Hi Carl,

>

>>> I would say a regular temperament is uniquely determined by a

>>> homomorphic mapping from the p-limit, or possibly another

>>> finitely-generated subgroup of the positive rationals, to an

>>> abstract free group of smaller rank. This can be specified by

>>> giving a wedgie, a kernel, or an explicit mapping.

>>

>>To many people on this list, like me, who haven't recently used or

>>studied abstract algebra, category theory, geometric algebra, and

>>Gene-Ward-Smith-ese (for "wedgie"), the above is probably about as

>>meaningful as this:

>

>This really isn't so mysterious at all if one is accustomed

>to the mathematical terms that Gene is using here.

It was actually Dave who wrote that, though I found your

comments helpful.

-Carl

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

>

> This is an interesting point: the difference between "bug" and

> "superpelog" for instance is mainly in the mapping, and in the number of

> generators in a typical scale. "Superpelog" is basically a 14-note scale

> with the mapping [<1 2 1 3|, <0 -2 6 -1|], while the "bug" mapping [<1 2

> 3 3|, <0 -2 -3 -1| is only good up to 11 notes, and a 9-note scale is

> typical. But the generator / period ratio is very similar: 260.76 /

> 1206.55 for 7-limit superpelog, 260.26 / 1200.00 for 5-limit bug, and

> 254.90 / 1194.64 for 7-limit bug. So audibly they're pretty similar

> tunings; the difference is in which notes are used to approximate

> consonant intervals. This is similar to the use of two different

> approximate fifths in ET's like 23-ET or 64-ET.

Wow; I was asking Paul Erlich about this sort of thing, mostly regarding a way that

you could make an sort of "atlas" of temperaments based on generator size. Clearly

many of these things defined precisely (if not uniquely) by their mapping overlap

greatly. Today's obsession:

If these things aren't exclusive, with clear-cut borders between them, can there at

least be a range of reasonability? Take "7-limit meantone", for example (my GOD!

can't get by a day without mention of meantone! you're all obsessed! :-). I started out

wanting to keep 7/4 greater than 3/2 greater than 5/4. Reasonable? My supreme

7th-grade math skills yeild a range of 5/9oct to 2/3oct. But that looks a little crazy

because if you pretend 9-equal is meantone, yer major third is smaller than yer minor

third!

So I tried keeping the fifth so that 5/4 > 6/5 > 7/6, and that seems to limit the fifth

to 4/7oct < g < 7/12oct. At one end, major and minor third become neutral, whilst

at the other, minor and subminor third are squinched into one. I don't see how these

can be objectively extended to other limits or how they could be calculated for TOP

tunings.

My rationale: if some dude prefers "bug" over "superpelog," what's to stop him from

using grossly-warped-yet-still-somehow-functional meantone?

> Another way to categorize tunings is by putting them on a branch of the

> scale tree (by their generator / period ratio if the period is around an

> octave). This would lump together scales that have different mappings,

> but sound roughly the same.

Herman I like this approach but still don't get this "scale tree." I tried to grow one of

my own but fell off and broke my wrist. Can anyone do a ground-up explanation? Or

an Encyclopaedia article?

Godspeed,

Jacob

on 8/15/04 7:22 PM, Dave Keenan <d.keenan@bigpond.net.au> wrote:

>> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

>>

>>> Maybe I can start with a question. What uniquely determines a

>> temperament?

>

> This is an excellent question.

>

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

> wrote:

>> I would say a regular temperament is uniquely determined by a

>> homomorphic mapping from the p-limit, or possibly another

>> finitely-generated subgroup of the positive rationals, to an

> abstract

>> free group of smaller rank. This can be specified by giving a

> wedgie,

>> a kernel, or an explicit mapping.

> But something we haven't agreed on is the following recasting of

> Kurt's question: "How can we tell, by listening or measuring,

> whether two tunings are different temperaments, or are merely

> slightly different tunings of the same temperament?"

Actually, yes this is quite a recasting. Because in fact my original

question was geared towards understanding the more technical sense of the

word "temperament" as it is often used on this list by Gene and others.

But I appreciate Dave's attempt here, and from its apparent tone I took it

as a rather playful exploration and inquiry into the possibility of a more

physical approach, rather than the definitive statement that Gene seemed to

hear. But Dave kind of asked for it with his Gene-mumbo-jumbo paragraph!

And yes, its true, I don't yet have what it takes to fully understand what

Gene said, so this set me back on my original quest a little. But I think

based on various clarifications that have come along, I can now fairly

safely recast Gene's answer to say that for most purposes on this list a

temperament is uniquely determined by the commas that are tempered out.

This is the confirmation that I was looking for. It confirms my

understanding of the way a temperament (as often used here and distinct from

the traditional common usage) is different from a tuning and may map to more

than one different tuning. I think I have heard this or something similar

echoed by several people now in this or other recent threads.

So if Gene ok's this, then I think maybe I can get back to my original

questioning that might shed more light on some of the gray areas that exist

for me along the polarity from temperament to tuning.

-Kurt

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:

> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> If these things aren't exclusive, with clear-cut borders between

them, can there at

> least be a range of reasonability? Take "7-limit meantone", for

example (my GOD!

> can't get by a day without mention of meantone! you're all obsessed!

:-). I started out

> wanting to keep 7/4 greater than 3/2 greater than 5/4. Reasonable?

My supreme

> 7th-grade math skills yeild a range of 5/9oct to 2/3oct. But that

looks a little crazy

> because if you pretend 9-equal is meantone, yer major third is

smaller than yer minor

> third!

>

> So I tried keeping the fifth so that 5/4 > 6/5 > 7/6, and that seems

to limit the fifth

> to 4/7oct < g < 7/12oct. At one end, major and minor third become

neutral, whilst

> at the other, minor and subminor third are squinched into one.

Sounds reasonable to me. You can relate this to the chroma thread:

25/24 is a chroma for a 7-note meantone MOS, and adopting it as a

comma turns the meantone MOS into 7-equal and squashes 5/4 and 6/5

together; 36/35 (squishing 6/5 and 7/6 together) is a 12-note meantone

chroma and works the same way with 12-et.

I don't see how these

> can be objectively extended to other limits or how they could be

calculated for TOP

> tunings.

I'm not quite sure what the question is, but other meantone chromas

which identify two 7-limit consonances are out there. Aside from

squashing 7/6 and 6/5 together, 12 also squashes 7/5 and 10/7

together, giving 50/49. 19 squashes 7/6 and 8/7 together; if you

regarded that as illegitimate you might want to put the boundries at

19 and 12. This has the advantage of giving an explicit criterion--you

are requiring that the meantone tuning not identify any two 7-limit

consonances nor put them in the wrong order, and this will be true

only if the tuning is somewhere between 19 and 12. This whole plan

pretty well breaks down when we get to the 11-limit, where 31, the

quintessential meantone tuning, identifies 11-limit consonances.

> Herman I like this approach but still don't get this "scale tree."

I tried to grow one of

> my own but fell off and broke my wrist. Can anyone do a ground-up

explanation? Or

> an Encyclopaedia article?

I think the reference was to the Stern-Brocot tree, which is closely

related to the Farey sequence.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>

> > Gene may have something to say about using tunings to categorize

> > temperaments -- he tried it initially but lately has been headed

> > toward using comma sequences. Gene, can you refresh us on the

> > gotchas of the tunings-based approach?

>

> I was suggesting both of these in connection to family

relationships.

> Which 7-limit temperament, if any, should we simply

> call "meantone"?

none. the meantone family was clearly designed to

represent 5-limit harmony, and while some of the

intervals of various versions of meantone resemble

7-limit or higher-limit ratios, it was rarely

consciously used that way historically, the only

real exception being the enthusiasm of composers

to use "augmented-6th" chords (which resemble 4:5:6:7

tetrads closely in some meantones).

> Is there an 11-limit temperament we might simply call

> "meantone" also?

again, i think it's not a good idea.

BTW ... Paul pointed out to me that Huygens only

wrote about meantone as representing up to the 7-limit.

his name should be associated with one of those

rather than with the 11-limit version which bears

his name now.

> My suggestion was to sort it out, to the extent possible,

> excluding anything too awful and then using TOP tuning.

you mean "using TOP" to categorize? can you elaborate?

-monz

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

But I think

> based on various clarifications that have come along, I can now fairly

> safely recast Gene's answer to say that for most purposes on this list a

> temperament is uniquely determined by the commas that are tempered out.

This is precisely what I meant, though I restricted that to regular

temperaments. Actually, I thought it was also what I said.

Irregular temperaments are tempered *scales*, and so are different

beasties anyway.

hi Jacob,

i know you're talking about 7-limit meantone here,

but i thought i'd toss this in about 5-limit:

the limits of ET 5-limit meantones are generally

considered to be 12-ET at one extreme and 19-ET

at the other.

the main reason for this is the size of the two

different semitones which occur in all meantones

except 12-ET: the diatonic (as A:Bb) and chromatic

(as A:A#). in 12-ET the difference disappears and

these are actually the same.

in all other meantones the generator is smaller

than 2^(7/12) = 700 cents, and the chromatic semitone

is smaller than the diatonic.

narrowing the generator all the way down to

2^(11/19) = 694 + 14/19 cents, results in an

exact division of the 19edo "whole-tone"

(189 + 9/19 cents) into 3 parts. thus, the

diatonic semitone is exactly twice as large

as the chromatic semitone.

narrowing the generator to a size smaller than

2^(11/19) results in tunings which seem strange

for music composed with meantone in mind, because

the enharmonic difference (between, say, A# and Bb)

becomes larger than the difference between a

nominal and its accidentaled relative.

in the other direction, extending the generator

to a size larger than 2^(7/12) makes the chromatic

semitones larger than the diatonic, which resembles

pythagorean tuning, and becomes nearly identical

to pythagorean at 2^(31/53) = 701 + 47/53 cents.

-monz

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:

> Wow; I was asking Paul Erlich about this sort of thing,

> mostly regarding a way that you could make an sort of

> "atlas" of temperaments based on generator size. Clearly

> many of these things defined precisely (if not uniquely)

> by their mapping overlap greatly. Today's obsession:

>

> If these things aren't exclusive, with clear-cut borders

> between them, can there at least be a range of reasonability?

> Take "7-limit meantone", for example (my GOD! can't get by

> a day without mention of meantone! you're all obsessed! :-).

> I started out wanting to keep 7/4 greater than 3/2 greater

> than 5/4. Reasonable? My supreme 7th-grade math skills

> yeild a range of 5/9oct to 2/3oct. But that looks a

> little crazy because if you pretend 9-equal is meantone,

> yer major third is smaller than yer minor third!

>

> <etc. -- snip>

oops ... i had meant to include this, which makes

it all plain in a very visual way:

http://tonalsoft.com/enc/index2.htm?../monzo/meantone/cycles.htm

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> hi Jacob,

>

>

> i know you're talking about 7-limit meantone here,

> but i thought i'd toss this in about 5-limit:

>

>

> the limits of ET 5-limit meantones are generally

> considered to be 12-ET at one extreme and 19-ET

> at the other.

>

>

> the main reason for this is the size of the two

> different semitones which occur in all meantones

> except 12-ET: the diatonic (as A:Bb) and chromatic

> (as A:A#). in 12-ET the difference disappears and

> these are actually the same.

>

>

> in all other meantones the generator is smaller

> than 2^(7/12) = 700 cents, and the chromatic semitone

> is smaller than the diatonic.

>

>

> narrowing the generator all the way down to

> 2^(11/19) = 694 + 14/19 cents, results in an

> exact division of the 19edo "whole-tone"

> (189 + 9/19 cents) into 3 parts. thus, the

> diatonic semitone is exactly twice as large

> as the chromatic semitone.

>

>

> narrowing the generator to a size smaller than

> 2^(11/19) results in tunings which seem strange

> for music composed with meantone in mind, because

> the enharmonic difference (between, say, A# and Bb)

> becomes larger than the difference between a

> nominal and its accidentaled relative.

>

>

> in the other direction, extending the generator

> to a size larger than 2^(7/12) makes the chromatic

> semitones larger than the diatonic, which resembles

> pythagorean tuning, and becomes nearly identical

> to pythagorean at 2^(31/53) = 701 + 47/53 cents.

>

>

>

> -monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> none. the meantone family was clearly designed to

> represent 5-limit harmony, and while some of the

> intervals of various versions of meantone resemble

> 7-limit or higher-limit ratios, it was rarely

> consciously used that way historically, the only

> real exception being the enthusiasm of composers

> to use "augmented-6th" chords (which resemble 4:5:6:7

> tetrads closely in some meantones).

On the one hand we have Dave, who thinks it all devolves to the actual

tuning. On the other hand we have you, who think that the precise same

tuning, and a consistent tuning map, doesn't mean diddly. This is

using history as a hindrence rather than a help, isn't it?

For certain, we can't very well adopt both points of view!

> BTW ... Paul pointed out to me that Huygens only

> wrote about meantone as representing up to the 7-limit.

> his name should be associated with one of those

> rather than with the 11-limit version which bears

> his name now.

If we are going to be picky, he wrote about 31-equal.

> you mean "using TOP" to categorize? can you elaborate?

7-limit meantone, which you say isn't meantone, has the exact same TOP

tuning as 5-limit meantone. So we have two temperaments, which have

exactly the same tuning and are being used in an entirely consistent

way, but giving them the same name brings down the wrath of the tuning

Inquisition. Let us by all means not call them by the same name, so as

to keep the fact that they have the same tuning hidden.

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> I can now fairly safely recast Gene's answer to say that

> for most purposes on this list a temperament is uniquely

> determined by the commas that are tempered out. This is

> the confirmation that I was looking for. It confirms my

> understanding of the way a temperament (as often used here

> and distinct from the traditional common usage) is different

> from a tuning and may map to more than one different tuning.

> I think I have heard this or something similar echoed by

> several people now in this or other recent threads.

i haven't really been sure where you going with this,

but here's my 2 cents:

it seems to me you're talking about the way certain EDOs,

for example, can be members of many different temperament

families, i.e.:

- 12edo belongs to meantone, schismic, augmented (diesic), etc.,

- 19edo belongs to meantone, magic, and kleismic

- 31edo belongs to meantone, miracle

some of these classifications can be seen in the very

bottom applet on my "bingo" page:

http://tonalsoft.com/enc/index2.htm?bingo.htm

is that what you're getting at?

-monz

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

>

> > none. the meantone family was clearly designed to

> > represent 5-limit harmony, and while some of the

> > intervals of various versions of meantone resemble

> > 7-limit or higher-limit ratios, it was rarely

> > consciously used that way historically, the only

> > real exception being the enthusiasm of composers

> > to use "augmented-6th" chords (which resemble 4:5:6:7

> > tetrads closely in some meantones).

>

> On the one hand we have Dave, who thinks it all devolves

> to the actual tuning. On the other hand we have you, who

> think that the precise same tuning, and a consistent tuning

> map, doesn't mean diddly. This is using history as a

> hindrence rather than a help, isn't it?

>

> For certain, we can't very well adopt both points of view!

>

> > BTW ... Paul pointed out to me that Huygens only

> > wrote about meantone as representing up to the 7-limit.

> > his name should be associated with one of those

> > rather than with the 11-limit version which bears

> > his name now.

>

> If we are going to be picky, he wrote about 31-equal.

if we're really going to be picky, he wrote about

how 31-equal comes so close to 1/4-comma meantone.

in fact, it's very interesting to me to see Huygens

refer to 31edo as "Division of the Octave into 31

Equal Parts", but to call 1/4-comma meantone

"temperament ordinaire" [ordinary temperament] or

often simply just plain "temperament".

> > you mean "using TOP" to categorize? can you elaborate?

>

> 7-limit meantone, which you say isn't meantone, has the

> exact same TOP tuning as 5-limit meantone. So we have two

> temperaments, which have exactly the same tuning and are

> being used in an entirely consistent way, but giving them

> the same name brings down the wrath of the tuning

> Inquisition. Let us by all means not call them by the

> same name, so as to keep the fact that they have the same

> tuning hidden.

wow, quit snarling for a while, would you? i merely

gave an opinion which you solicited and now all of a

sudden you mention the "tuning Inquisition" with what

sounds to me like a clear reference to me.

i think your wrath is simply the result of misunderstanding

what i meant, which i didn't express clearly.

of course, these tunings all belong to the meantone

family ... i never said that in my post, so it's clear

why you didn't get that from what i wrote.

i just meant that i think plain-old "meantone" should

be reserved for 5-limit, and then "septimal meantone"

or "huygens" for 7-limit, and whatever appropriate name

for 11-limit, etc.

certainly, i want to see Aunt Martha, Illegitimate

Uncle Simon, and all the other long-lost cousins

of the (perhaps somewhat dysfunctional?) meantone family.

... i can't wait to see what happens when they

get into arguments! ... hmm ... what a great

idea for a polymicrotonal opera ...

-monz

on 8/16/04 2:02 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> But I think

>> based on various clarifications that have come along, I can now fairly

>> safely recast Gene's answer to say that for most purposes on this list a

>> temperament is uniquely determined by the commas that are tempered out.

>

> This is precisely what I meant, though I restricted that to regular

> temperaments. Actually, I thought it was also what I said.

I'll reply to this in the original thread shortly.

> Irregular temperaments are tempered *scales*, and so are different

> beasties anyway.

So is what you are calling "tempered scales" closer to the historical

meaning of "temperament"?

Prior to physics-based tuning theory, everyone was talking about scales,

right? At that time people didn't know about frequency ratios per-se,

although length-of-string ratios are much older. On the other hand there

was clearly some sense even then that exact intervals were being tempered,

in spite of the lack of physical theory defining exact intervals.

Nonetheless it all dealt with application to scales and tunings.

My sense is that "temperament" and "tempered scale" used to be synonymous,

though I'm not quite sure when the term "temperament" was invented. I'm

presuming it was well before "well-temperament".

Regardless of history: what exactly is your distinction between

"temperament" and "tempered scale" and how does a "tempered scale" relate to

a tuning? Is this a generally agreed distinction, a distinction unique to

this community, or a more universal one?

-Kurt

>Prior to physics-based tuning theory, everyone was talking

>about scales, right?

Temperaments have been abstract entities apart from scales

since I joined this list in 1997, though Gene made the

distinction more precise and has stressed its importance.

I didn't fully grok it until Gene came along.

>Regardless of history: what exactly is your distinction between

>"temperament" and "tempered scale" and how does a "tempered scale"

>relate to a tuning? Is this a generally agreed distinction, a

>distinction unique to this community, or a more universal one?

I'll let Gene answer this, but note that there is also such a

thing as a scale (apart from a tempered scale) and it's importance

is something I have stressed in our discussions. But I can't find

Gene's definition just now, in the Encyclopaedia or on his site.

-Carl

on 8/16/04 2:02 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> But I think

>> based on various clarifications that have come along, I can now fairly

>> safely recast Gene's answer to say that for most purposes on this list a

>> temperament is uniquely determined by the commas that are tempered out.

>

> This is precisely what I meant, though I restricted that to regular

> temperaments. Actually, I thought it was also what I said.

Yes, basically but you gave more options including an explicit mapping, and

you used the term "kernel" instead of "comma".

But in any case I can return to my original line of questioning now, which

should clarify to Monz what I was up to. Actually I've already learned so

much that returning to my original line of questions feels slightly

artificial, but I'll try to do it anyway since otherwise I've kept all the

intervening thoughts to myself.

Lets look at starting with the 5-limit and tempering out the 225/224 to make

the 225:128 into an exact 7:4. So my understanding is no matter how you get

rid of the 225/224 comma starting from the 5-limit, you have the same

temperament as a result.

So let's look at at making a tuning in this temperament. One approach that

comes to mind is to alter the 3:2 an the 5:4 by the same factor in order to

make the 7:4 exact. So this involves applying the ratio (225/224)^(1/4) to

3:2 and to 5:4. This value (225/224)^(1/4) is something like a comma

itself, though I guess it isn't standard use of the term. But this is one

of those irrational comma-like quantities that I had talked about. It is a

distributed granule of a comma. Now I understand this is not called a

comma. Originally I thought of tempering as bending the lattice in various

ways and I thought of the ratios involved in the bending as being some kind

of comma. And I thought of the different ways of altering the lattice in

order to achieve the different tempered tunings as different temperaments,

which were to me *structurally* different even though the same comma was

being tempered out. This might be clearer if I give an example of a

different tempered tuning from the same "temperament" that to me is

structurally different.

So let's look at another example. One thought about tempering out 225/224

is that if 4:7 is to be made exact, why not make 5:7 or 6:7 exact too. One

or the other has to be picked; you can't have both by tweaking a lattice

interval in all occurrences (which is what I'm looking at for the moment).

So let's say we want to make 5:7 exact at the expense of 6:7. This allows

you to have an exact 4:5:7 chord. Now this means *not* tweaking the 5:4

lattice interval, but instead tweaking only the 3:2, by the square of the

ratio that was used in the previous example. Now to me this is structurally

different because it involves a different assumption of what the goal of the

tempering process is. Does that make sense? Yet the result is considered

the same temperament. Yet I guess Gene would say this is all working in a

real number space rather than an abelian group, and so it is just a matter

or tuning and not of defining a temperament.

So this pretty much finishes my original line of thinking (or the part of it

that I remember) which I think reveals most of the confusion I was having

about "what is a temperament". At this moment I have no specific questions

but will be interested in any responses to the above lines of thinking. I

hope I stated it clearly enough.

-Kurt

Hi Kurt (and Gene)!

Sorry to keep butting in here, just a few points:

>Yes, basically but you gave more options including an explicit

>mapping, and you used the term "kernel" instead of "comma".

A kernel is actually the set of all the commas that vanish

in a temperament -- which I think also includes things like

powers of vanishing commas, and therefore contains an infinite

number of commas.

>Lets look at starting with the 5-limit and tempering out the

>225/224 to make the 225:128 into an exact 7:4. So my

>understanding is no matter how you get rid of the 225/224

>comma starting from the 5-limit, you have the same temperament

>as a result.

Strictly speaking, 225:224 is a 7-limit interval, so you

can't temper it out of the 5-limit lattice.

>So let's look at at making a tuning in this temperament.

Just to refresh, a tuning is an infinite thing. Temperaments

are tunings, but scales are not. Sometimes scales can be

explained in terms of tunings, though, as Gene recently did

in his "Reverse engineering a scale" thread.

>One approach that comes to mind is to alter the 3:2 an the

>5:4 by the same factor in order to make the 7:4 exact. So

>this involves applying the ratio (225/224)^(1/4) to 3:2 and

>to 5:4. This value (225/224)^(1/4) is something like a comma

>itself, though I guess it isn't standard use of the term.

>But this is one of those irrational comma-like quantities

>that I had talked about.

It's an "error". That's what we'd call it. Sorry I couldn't

think of that the other night!

...That's all for now. Let me hint again that your "original

line of questioning" is getting you close to rediscovering TOP

tuning. OTOH, we were close for years around here until Paul

put it all together. So if you want to cut to the chase, I

can come over and explain it sometime.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> I'll let Gene answer this, but note that there is also such a

> thing as a scale (apart from a tempered scale) and it's importance

> is something I have stressed in our discussions. But I can't find

> Gene's definition just now, in the Encyclopaedia or on his site.

That definition was much hated. What about this:

A discrete set of real numbers including 0 giving the value, in cents,

of frequency ratios relative to the base frequency represented by 0. A

periodic scale has an indexing mapping s from the integers and a

positive integer n such that s[i+n]-s[i] is a constant for all i. A

finite scale is a finite set of cents values, including 0. Scales can

also be given multipliciatively, including using rational numbers

only, so long as the corresponding values in cents satisfy the

requirements for being a scale.

Note that using this definition, the p-limit for any odd prime p is

not a scale, but any equal division of the octave will be.

>> I'll let Gene answer this, but note that there is also such a

>> thing as a scale (apart from a tempered scale) and it's importance

>> is something I have stressed in our discussions. But I can't find

>> Gene's definition just now, in the Encyclopaedia or on his site.

>

>That definition was much hated.

I remember. I wish I had it to look back on now. It's probably

here in my inbox. Any terms other than "scale" it might have

contained?

>What about this:

>

>A discrete set of real numbers including 0 giving the value, in

>cents, of frequency ratios relative to the base frequency

>represented by 0.

This sounds imprecise. In particular, it sounds like we are

being restricted to rational numbers.

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Actually, yes this is quite a recasting. Because in fact my

original

> question was geared towards understanding the more technical sense

of the

> word "temperament" as it is often used on this list by Gene and

others.

>

> But I appreciate Dave's attempt here, and from its apparent tone I

took it

> as a rather playful exploration and inquiry into the possibility

of a more

> physical approach, rather than the definitive statement that Gene

seemed to

> hear. But Dave kind of asked for it with his Gene-mumbo-jumbo

paragraph!

>

Yes I did, didn't I. :-)

I wanted to make the point, in a humorous way, that this isn't the

tuning-math list, and it was quite possible to say the same thing in

a way that you, and many others, would have immediately understood,

as others have subsequently done.

Then I wanted to remind us of the possibility of an extreme

experiential point of view as balance to Gene's extreme mathematical

one.

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >A discrete set of real numbers including 0 giving the value, in

> >cents, of frequency ratios relative to the base frequency

> >represented by 0.

>

> This sounds imprecise. In particular, it sounds like we are

> being restricted to rational numbers.

Only if you think a ratio has to be a ratio of integers; but this is

not the case. The opposite viewpoint goes back to ancient Greece and

has been accepted ever since; if you say the ratio of the side of a

square to its diagonal cannot be a ratio of integers, which the Greeks

were willing to say, you admit ratios need not be commensureable; as

Euclid puts it "two magnitudes are commensurable if and only if their

ratio is the ratio of a number to a number", where of course for

"magnitude" we would say "positive real number" and for "number" we

would say "positive integer".

Now, probably we don't want to refer people to Euclid in order to sort

this all out, so what would be good at this point is a suggestion for

saying it in a way which would be less confusing.

on 8/16/04 5:48 PM, Carl Lumma <ekin@lumma.org> wrote:

>

> Hi Kurt (and Gene)!

>

> Sorry to keep butting in here, just a few points:

>

>> Yes, basically but you gave more options including an explicit

>> mapping, and you used the term "kernel" instead of "comma".

>

> A kernel is actually the set of all the commas that vanish

> in a temperament -- which I think also includes things like

> powers of vanishing commas, and therefore contains an infinite

> number of commas.

Yes, quite frankly its all the same to me. Maybe the kernel then is the

"expansion" of the commas that are involved into an entire *mapping*? Just

taking a guess. It sounded like like a mapping makes everything explicit

and doesn't require a list of commas tempered out, and is a more general

entity. Yet Gene didn't equate kernel to mapping, so mapbe kernel is still

less general.

>> Lets look at starting with the 5-limit and tempering out the

>> 225/224 to make the 225:128 into an exact 7:4. So my

>> understanding is no matter how you get rid of the 225/224

>> comma starting from the 5-limit, you have the same temperament

>> as a result.

>

> Strictly speaking, 225:224 is a 7-limit interval, so you

> can't temper it out of the 5-limit lattice.

Yes, yes, I keep forgetting. The way I naturally think keeps getting in the

way. I think: "Ok, I'm in the 5-limit. Gee here's a good approximation to

4:7. Let's make it exact." So slopily it looked to me like I was tempering

the 5 limit. And damn it, I *was* tempering the 5 limit. You math heads

get out of here. You're obscuring the obvious. ;) See why math is a

problem? It makes us retroactively unconscious. ;)

>> So let's look at at making a tuning in this temperament.

>

> Just to refresh, a tuning is an infinite thing.

Ok, I was using tuning as a finite thing, because damn it if you try to tune

an infinite thing you won't finish tuning it before you die. You math heads

don't know what "tuning" really means, now do you. Get off your computers

and touch something real. ;)

> Temperaments

> are tunings, but scales are not. Sometimes scales can be

> explained in terms of tunings, though, as Gene recently did

> in his "Reverse engineering a scale" thread.

>

>> One approach that comes to mind is to alter the 3:2 an the

>> 5:4 by the same factor in order to make the 7:4 exact. So

>> this involves applying the ratio (225/224)^(1/4) to 3:2 and

>> to 5:4. This value (225/224)^(1/4) is something like a comma

>> itself, though I guess it isn't standard use of the term.

>> But this is one of those irrational comma-like quantities

>> that I had talked about.

>

> It's an "error". That's what we'd call it. Sorry I couldn't

> think of that the other night!

I think you might have, it sounds really familiar. But I didn't like the

word and don't like it now. I know what you mean, but to me I was

correcting an error in the 4:7, so that was the distributed correction.

I can learn to think like you and I have already assimilated a lot of

abstract stuff that now causes me to have problems communicating with

ordinary people. Really! ;)

> ...That's all for now. Let me hint again that your "original

> line of questioning" is getting you close to rediscovering TOP

> tuning. OTOH, we were close for years around here until Paul

> put it all together. So if you want to cut to the chase, I

> can come over and explain it sometime.

Sure. My intuition is telling me that TOP becomes a way of classifying

tempered scales.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> So let's look at at making a tuning in this temperament. One

approach that

> comes to mind is to alter the 3:2 an the 5:4 by the same factor in

order to

> make the 7:4 exact. So this involves applying the ratio

(225/224)^(1/4) to

> 3:2 and to 5:4. This value (225/224)^(1/4) is something like a comma

> itself, though I guess it isn't standard use of the term.

Strange you should mention this; I just posted something for Monz over

on tuning-math which mentions this tuning (which I call 1/4-kleismic,

in analogy to 1/4-comma.)

Now to me this is structurally

> different because it involves a different assumption of what the

goal of the

> tempering process is. Does that make sense?

It's the reverse of what I would mean by "structurally different";

we've got exactly the same structure, and variations in tuning which

are, after all, not that great.

Yet the result is considered

> the same temperament. Yet I guess Gene would say this is all

working in a

> real number space rather than an abelian group, and so it is just a

matter

> or tuning and not of defining a temperament.

I would say it is working with the tuning map and not the temperament

map. Tuning "factors through" the temperament, as mathematicians are

wont to say (before they start drawing diagrams with lots of arrows,

so I'd better stop now.)

>> A kernel is actually the set of all the commas that vanish

>> in a temperament -- which I think also includes things like

>> powers of vanishing commas, and therefore contains an infinite

>> number of commas.

>

>Yes, quite frankly its all the same to me. Maybe the kernel then

>is the "expansion" of the commas that are involved into an entire

>*mapping*? Just taking a guess. It sounded like like a mapping

>makes everything explicit and doesn't require a list of commas

>tempered out, and is a more general entity. Yet Gene didn't equate

>kernel to mapping, so mapbe kernel is still less general.

Well, I don't know much about kernels. Just learned about them

the other day. I found the mathworld definition helpful. Around

here, the word has usually occurred in a phrase like: 'Is this

interval in the tuning's kernel?' This phrase just means: 'Does

this interval vanish in the tuning?'

>>> Lets look at starting with the 5-limit and tempering out the

>>> 225/224 to make the 225:128 into an exact 7:4. So my

>>> understanding is no matter how you get rid of the 225/224

>>> comma starting from the 5-limit, you have the same temperament

>>> as a result.

>>

>> Strictly speaking, 225:224 is a 7-limit interval, so you

>> can't temper it out of the 5-limit lattice.

>

>Yes, yes, I keep forgetting. The way I naturally think keeps

>getting in the way. I think: "Ok, I'm in the 5-limit. Gee

>here's a good approximation to 4:7. Let's make it exact." So

>slopily it looked to me like I was tempering the 5 limit. And

>damn it, I *was* tempering the 5 limit.

You're tempering 5-limit intervals, but saying 'let's make it

an exact 4:7' implies a 7-limit universe.

>You math heads get out of here. You're obscuring the obvious. ;)

>See why math is a problem? It makes us retroactively unconscious. ;)

Well I never! You're the one who taught me about integrals!

>>> So let's look at at making a tuning in this temperament.

>>

>> Just to refresh, a tuning is an infinite thing.

>

>Ok, I was using tuning as a finite thing, because damn it if you

>try to tune an infinite thing you won't finish tuning it before

>you die. You math heads don't know what "tuning" really means,

>now do you. Get off your computers and touch something real. ;)

I've been bitten by using the word "temperament" to mean scale

several times in the last year or so. In one such case I

retorted that infinite things don't matter in music theory. But

in fact they do.

>> Temperaments

>> are tunings, but scales are not. Sometimes scales can be

>> explained in terms of tunings, though, as Gene recently did

>> in his "Reverse engineering a scale" thread.

>>

>>> One approach that comes to mind is to alter the 3:2 an the

>>> 5:4 by the same factor in order to make the 7:4 exact. So

>>> this involves applying the ratio (225/224)^(1/4) to 3:2 and

>>> to 5:4. This value (225/224)^(1/4) is something like a comma

>>> itself, though I guess it isn't standard use of the term.

>>> But this is one of those irrational comma-like quantities

>>> that I had talked about.

>>

>> It's an "error". That's what we'd call it. Sorry I couldn't

>> think of that the other night!

>

>I think you might have, it sounds really familiar. But I didn't

>like the word and don't like it now. I know what you mean, but

>to me I was correcting an error in the 4:7, so that was the

>distributed correction.

From the JI point of view, there is no error in 4:7. You're

putting error into the 3 and/or 5 axis to get 225/128 = 7/4.

>> ...That's all for now. Let me hint again that your "original

>> line of questioning" is getting you close to rediscovering TOP

>> tuning. OTOH, we were close for years around here until Paul

>> put it all together. So if you want to cut to the chase, I

>> can come over and explain it sometime.

>

>Sure. My intuition is telling me that TOP becomes a way of

>classifying tempered scales.

Well, Gene suggested it be used to define the canonical tuning

for any temperament, and this idea seems to have met with a

cautious quasi-consensus, which is the best kind of consensus

ever likely to emerge from the tuning-math list. :)

Once you have a way of finding a canonical tuning for any

temperament, you can try to put temperaments into families

based on the tunings (and ultimately scales) they give rise

to. Another approach is to base the families on comma

sequences. I'm not clear on the pros/cons of these approaches.

-Carl

>> Yet the result is considered

>> the same temperament. Yet I guess Gene would say this is all

>> working in a real number space rather than an abelian group,

>> and so it is just a matter or tuning and not of defining a

>> temperament.

>

>I would say it is working with the tuning map and not the temperament

>map. Tuning "factors through" the temperament, as mathematicians are

>wont to say (before they start drawing diagrams with lots of arrows,

>so I'd better stop now.)

Another way of saying it is that you choose a mapping (temperament)

and an error function, calculate a tuning by minimizing the error

given by your function and mapping, and finally produce a scale by

choosing a reasonable number of pitches from that tuning.

Does this help?

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> So let's say we want to make 5:7 exact at the expense of 6:7. This

allows

> you to have an exact 4:5:7 chord. Now this means *not* tweaking the 5:4

> lattice interval, but instead tweaking only the 3:2, by the square

of the

> ratio that was used in the previous example.

In other words, we flatten the fifth by sqrt(225/224), and leave 2, 5,

and 7 pure. This gives us a fifth we can use as a meantone fifth--in

fact, something pretty close to the fifth of 55-equal, which should

make Monz happy. It is 2*sqrt(14)/5, or 698.099 cents in size, and it

gives us a meantone third sharp by 3136/3125, or 6.083 cents.

Meanwhile, we also have a pure third; we've got an inconsistent system

cooking here, with two kinds of major third. Find an intelligent way

of putting together two or more chains of 1/2-kleismic meantone fifths

separated by 5/4, 7/4, or 7/5, and you're in business. A similar idea

would be to use the 1/2-kleismic major third as a magic temperament

generator.

on 8/16/04 7:27 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> Yet the result is considered

>>> the same temperament. Yet I guess Gene would say this is all

>>> working in a real number space rather than an abelian group,

>>> and so it is just a matter or tuning and not of defining a

>>> temperament.

>>

>> I would say it is working with the tuning map and not the temperament

>> map. Tuning "factors through" the temperament, as mathematicians are

>> wont to say (before they start drawing diagrams with lots of arrows,

>> so I'd better stop now.)

>

> Another way of saying it is that you choose a mapping (temperament)

> and an error function, calculate a tuning by minimizing the error

> given by your function and mapping, and finally produce a scale by

> choosing a reasonable number of pitches from that tuning.

>

> Does this help?

Yes, this brings it into another familiar territory since in fact I have

thought about and asked questions in the past about optimizing ...umm...

scales, I think. And in fact TOP and standard-octave TOP variants came up

at that time. This leaves each pitch free to be optimized independently.

In my examples I was giving it more "structure", i.e. tuning an entire

lattice at once, perhaps you would say by scaling it. I wanted to

distinguish that special-case from the general case of tuning each pitch

independently, but I can see how that is probably not important enough to

create specific mathematical structures for, because there are too many

choices available to be worth categorizing. As Gene said the various

tunings don't differ by much. In fact my special cases fit into the general

case simply by stating what specifically I want to optimize, e.g. the 4:7

and the 4:5 with no consideration of the 4:6. That's coming at it from the

other end though, and is not the most natural way for me to think about it.

I don't like applying generality and limiting it if instead I can apply less

generality. But that's just me, and probably not (for example) Paul, who

I'll bet likes to apply the most generality possible and then apply

constraints to it.

-Kurt

Gene Ward Smith wrote:

> I think the reference was to the Stern-Brocot tree, which is closely

> related to the Farey sequence.

> > http://mathworld.wolfram.com/Stern-BrocotTree.html

> > http://mathworld.wolfram.com/FareySequence.html

Specifically, the reference is to Erv Wilson's application of the Stern-Brocot tree for classifying the structures of what he calls "MOS" scales (which stands for "moments of symmetry"). The fractions on the scale tree refer to the relative size of the scale's generator; different sizes of generator result in different scale structures.

http://www.anaphoria.com/sctree.PDF

http://www.anaphoria.com/hrgm.PDF

http://www.anaphoria.com/line.PDF

These are all available on the Wilson Archives web site:

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> > >>Another way to categorize tunings is by putting them on a branch of the >>scale tree (by their generator / period ratio if the period is

> > around an > >>octave).

> > > To actually use this to categorize you'd have to use a standard

> version of the generator, such as the 1 < g < sqrt(period).

Note that I wrote "tunings", not "temperaments". Categorizing tunings by their position on the scale tree associates tunings that have the same scale structure (in numbers of small and large steps). Any particular tuning, with a particular pattern of MOS scales that can be displayed on a horagram, is compatible with a number of different temperaments, although usually only a few make sense. Similarly, any particular temperament can be tuned in different ways, producing different scale structures. Both of these categorizations are useful in different ways.

Take for example Wilson's golden horagram #9, which is labeled "Hanson". This has a generator of (4 phi + 1) / (15 phi + 4), or approximately 317.17 cents. One mapping that makes sense for this tuning is [<1, 0, 1, -3|, <0, 6, 5, 22|], which is a temperament for which the name "Hanson" has been suggested (it's the one that best fits the 14/53 generator implied in Larry Hanson's paper). But for more accuracy you can use the alternative mapping [<1, 0, 1, 11|, <0, 6, 5, -31|]. Or if you can tolerate a larger error, [<1, 0, 1, 2|, <0, 6, 5, 3|] will do. But however you interpret it, it's the same tuning, with the same scale structure.

> One > >>thing this could be useful for is notation: it would make sense to use >>the same notation for all scales with roughly the same g/p ratio (like >>meantone 504.13 / 1201.7 and flattone 507.14 / 1202.54, or dominant >>495.88 / 1195.23 and garibaldi 498.12 / 1200.76).

> > > Where do you switch? Why, for instance, don't you continue on to

> superpyth?

Where you switch depends on how many notes you use in the notation. If you're using 12 notes, the node in the scale tree adjacent to 5/12 is 7/17; the useful range of superpyth is on the other side of 7/17, so it's on a different branch. If you carry the generators out far enough, you'd use different notations for a version of meantone with a 504.13 / 1201.7 ratio and a version of flattone with a 507.14 / 1202.54 ratio.

>> Another way of saying it is that you choose a mapping (temperament)

>> and an error function, calculate a tuning by minimizing the error

>> given by your function and mapping, and finally produce a scale by

>> choosing a reasonable number of pitches from that tuning.

>>

>> Does this help?

>

>Yes, this brings it into another familiar territory since in fact I have

>thought about and asked questions in the past about optimizing ...umm...

>scales, I think. And in fact TOP and standard-octave TOP variants came up

>at that time. This leaves each pitch free to be optimized independently.

>

>In my examples I was giving it more "structure", i.e. tuning an entire

>lattice at once, perhaps you would say by scaling it.

Actually, this is exactly what TOP does -- tunes the whole lattice

by 'scaling' it.

>I wanted to distinguish that special-case from the general case of

>tuning each pitch independently,

As far as regular temperaments are concerned, one doesn't get to

tune the pitches independently. Tuning pitches independently gives

you what Gene would call a circulating temperament.

>but I can see how that is probably not important enough to

>create specific mathematical structures for, because there are too many

>choices available to be worth categorizing. As Gene said the various

>tunings don't differ by much. In fact my special cases fit into the

>general case simply by stating what specifically I want to optimize,

>e.g. the 4:7 and the 4:5 with no consideration of the 4:6.

Now you just need to specify how many dimensions you want your

temperament to have (or equivalently which commas you want to

vanish) and you can have what you want!

-Carl

hi Kurt,

not really saying much here to clarify your questions,

but i know a bit about the historical background ...

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Prior to physics-based tuning theory, everyone was

> talking about scales, right? At that time people

> didn't know about frequency ratios per-se, although

> length-of-string ratios are much older. On the other

> hand there was clearly some sense even then that exact

> intervals were being tempered, in spite of the lack of

> physical theory defining exact intervals. Nonetheless

> it all dealt with application to scales and tunings.

regarding tuning by measurement of string-lengths:

the earliest literature which definitely describes tuning

by measurement of string-length ratios, AFAIK, is by

Philolaus c.400s BC, and then a bit later and in much

more detail by Archytas, c.300s BC ... both ancient Greeks.

see my so far still very crude timeline of music-theory:

http://tonalsoft.com/enc/index2.htm?../monzo/timeline/timeline.htm

however, studying a Babylonian tablet from c.1800 BC,

i believe that i have deciphered a Sumerian method

for computing "pythagorean" tuning c.3000-2500 BC:

http://tonalsoft.com/enc/index2.htm?../monzo/sumerian/sumerian-tuning.

htm

OR

regarding temperament:

this is at least as old as ancient Greece, 300s BC.

Aristoxenus, perhaps the most profound of all the

ancient Greek music-theorists, steadfastly refused

to mention ratios at all in his treatise, preferring

instead to employ the new geometrical ideas that had

only recently been laid down by Euclid.

he discusses "tuning by concords" (successive

"perfect-4ths" and "perfect-5ths"), in such a way

that after finding 12 different notes, one still has

another "perfect-5th" between the two ends of the

chain, which thus makes it a circle and not a chain.

this cannot happen using acoustically "pure" pythagorean

tuning, and so it must necessarily indicate temperament.

see:

http://tonalsoft.com/enc/index2.htm?../monzo/aristoxenus/318tet.

htm&concords

OR

and scroll down a bit further to see some Excel graphs i

made of another examination of his "tuning by concords".

-monz

hi Kurt,

not really saying much here to clarify your questions,

but i know a bit about the historical background ...

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Prior to physics-based tuning theory, everyone was

> talking about scales, right? At that time people

> didn't know about frequency ratios per-se, although

> length-of-string ratios are much older. On the other

> hand there was clearly some sense even then that exact

> intervals were being tempered, in spite of the lack of

> physical theory defining exact intervals. Nonetheless

> it all dealt with application to scales and tunings.

regarding tuning by measurement of string-lengths:

the earliest literature which definitely describes tuning

by measurement of string-length ratios, AFAIK, is by

Philolaus c.400s BC, and then a bit later and in much

more detail by Archytas, c.300s BC ... both ancient Greeks.

see my so far still very crude timeline of music-theory:

http://tonalsoft.com/enc/index2.htm?../monzo/timeline/timeline.htm

however, studying a Babylonian tablet from c.1800 BC,

i believe that i have deciphered a Sumerian method

for computing "pythagorean" tuning c.3000-2500 BC:

http://tonalsoft.com/enc/index2.htm?../monzo/sumerian/sumerian-tuning.

htm

OR

regarding temperament:

this is at least as old as ancient Greece, 300s BC.

Aristoxenus, perhaps the most profound of all the

ancient Greek music-theorists, steadfastly refused

to mention ratios at all in his treatise, preferring

instead to employ the new geometrical ideas that had

only recently been laid down by Euclid.

he discusses "tuning by concords" (successive

"perfect-4ths" and "perfect-5ths"), in such a way

that after finding 12 different notes, one still has

another "perfect-5th" between the two ends of the

chain, which thus makes it a circle and not a chain.

this cannot happen using acoustically "pure" pythagorean

tuning, and so it must necessarily indicate temperament.

see:

http://tonalsoft.com/enc/index2.htm?../monzo/aristoxenus/318tet.

htm&concords

OR

and scroll down a bit further to see some Excel graphs i

made of another examination of his "tuning by concords".

-monz

--- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> Hi Kurt (and Gene)!

>

> Sorry to keep butting in here, just a few points:

>

> > Yes, basically but you gave more options including an explicit

> > mapping, and you used the term "kernel" instead of "comma".

>

> A kernel is actually the set of all the commas that vanish

> in a temperament -- which I think also includes things like

> powers of vanishing commas, and therefore contains an infinite

> number of commas.

yep.

each "vanishing comma" (vapro) is also a promo, which

is simply the ratio which vanishes in a temperament,

along with all of its powers, which also vanish.

all vapros are promos, but not all promos are vapros.

the distinction is that a promo need not vanish.

if a JI tuning is set up so that a certain "comma"

is considered a unison, it most definitely does not

vanish in JI, but if all of its powers are still also

considered unisons (as in standard periodicity-blocks),

then they all together constitute a promo.

in cases where powers of unison-vectors are *not*

considered to be unisons, as for example in torsional-blocks

like the Helmholtz-24 and Groven-36 schismic tunings, then

certain ones of those powers are simply unison-vectors

and not members of a promo, and the other powers

(such as the "comma" itself if one of the higher powers

is a unison-vector) are simply valid scale degrees.

-monz

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

>

> > I'll let Gene answer this, but note that there is also

> > such a thing as a scale (apart from a tempered scale) and

> > it's importance is something I have stressed in our

> > discussions. But I can't find Gene's definition just now,

> > in the Encyclopaedia or on his site.

>

> That definition was much hated. What about this:

>

> A discrete set of real numbers including 0 giving the

> value, in cents, of frequency ratios relative to the base

> frequency represented by 0. A periodic scale has an indexing

> mapping s from the integers and a positive integer n such

> that s[i+n]-s[i] is a constant for all i. A finite scale

> is a finite set of cents values, including 0. Scales can

> also be given multipliciatively, including using rational

> numbers only, so long as the corresponding values in cents

> satisfy the requirements for being a scale.

>

> Note that using this definition, the p-limit for any

> odd prime p is not a scale, but any equal division of

> the octave will be.

that seems like it might be a great definition for a

mathematician ... but for a musician, i think the first

thing that should be mentioned in a definition of "scale"

is that it is a collection of musical pitches.

sure, the set of reals *represents* those pitches,

but the actual physical scale *is* notes, not numbers.

-monz

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> As far as regular temperaments are concerned, one doesn't get to

> tune the pitches independently. Tuning pitches independently gives

> you what Gene would call a circulating temperament.

Only if it circulates!

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> sure, the set of reals *represents* those pitches,

> but the actual physical scale *is* notes, not numbers.

That means nothing in the Scala scale archive is a scale, since none

of them tell you what the notes are. Almost nothing which has been

posted on the tuning list over the last ten years, and called a scale,

actually is if we follow this definition.

>> As far as regular temperaments are concerned, one doesn't get to

>> tune the pitches independently. Tuning pitches independently gives

>> you what Gene would call a circulating temperament.

>

>Only if it circulates!

So whaddya call an unequal scale which is playable in all keys

but which doesn't have all the good keys lumped together on the

chain of fifths?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> As far as regular temperaments are concerned, one doesn't get to

> >> tune the pitches independently. Tuning pitches independently gives

> >> you what Gene would call a circulating temperament.

> >

> >Only if it circulates!

>

> So whaddya call an unequal scale which is playable in all keys

> but which doesn't have all the good keys lumped together on the

> chain of fifths?

I call it circulating. I think someone (Parizek?) suggested that

should be circular.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> in cases where powers of unison-vectors are *not*

> considered to be unisons, as for example in torsional-blocks

> like the Helmholtz-24 and Groven-36 schismic tunings, then

> certain ones of those powers are simply unison-vectors

> and not members of a promo, and the other powers

> (such as the "comma" itself if one of the higher powers

> is a unison-vector) are simply valid scale degrees.

more correctly i should have said:

... the other powers (such as the "comma" itself if

one of the higher powers is a unison-vector) are simply

valid degrees of the tuning-system.

then i thought to add this:

scales would be smaller subsets of both of these tunings,

but usually would not include a pair of notes separated

by the "comma", but only one of them.

... so at least in terms of deriving smaller subset scales

from these tuning-systems, the "comma" *does* become a sort

of unison-vector after all, which means that the "comma"

and all of its powers *are* in some sense considered

unisons, which means that they do in the end form a promo.

in terms of the temperament as a tuning-system, they

do not form a promo, because these tunings must be

classified as torsional-blocks and not regular

periodicity-blocks, and certain powers of the "comma"

are needed to distinguish the torsional divisions.

-monz

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

>

> > sure, the set of reals *represents* those pitches,

> > but the actual physical scale *is* notes, not numbers.

>

> That means nothing in the Scala scale archive is a scale,

> since none of them tell you what the notes are. Almost

> nothing which has been posted on the tuning list over the

> last ten years, and called a scale, actually is if we follow

> this definition.

well, ok, i sure can see that point.

-monz

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

/tuning/topicId_55471.html#55537

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > if i had somebody around who could have given me

> > answers like that 10 years ago, i would have saved

> > myself a ridiculous amount of time and effort.

>

> Yes Monz,

>

> As I've said many times, I don't know what we'd do without your

> encyclopedia. But don't you think that when an encyclopedia that's

> edited by one person starts having entries for new terms added at

> the whim of that editor and deleted again within a few days and new

> terms put in their place, it seriously undermines the authority of

> that encyclopedia. Why should anyone take any notice of any of it

if

> that sort of thing can happen?

***With my apologies to my friend "the Monz," I intend to agree with

Dave here. Monz is intrigued by "exotica"... see his historical

early-civilization websites... Quite frankly, I think "exotica"

should not be a part of terminology and rather than "colorful"

creates an alternate club... a kind of tuning speakeasy (easyspeak??)

J. Pehrson

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

>

> > sure, the set of reals *represents* those pitches,

> > but the actual physical scale *is* notes, not numbers.

>

> That means nothing in the Scala scale archive is a scale,

Yes. That is true, of course. Is that somehow suprising?

But of course it would be too tedious to say "representation of a

scale" all the time when everyone knows that's what they are (or at

least I _thought_ everyone knew), so we just call them scales. No

problem.

But when defining what a scale is, surely we want to mention that

it's an actual musical thing with notes (which can be represented in

many different ways), not merely the collection of numbers which

constitute one particular mathematical representation.

One shouldn't mistake the map for the territory, especially not a

single kind of map.

> since none

> of them tell you what the notes are.

But they do tell you what the notes are, once you know how the

representation works.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> But they do tell you what the notes are, once you know how the

> representation works.

No they don't. You need to specify how many Hertz 1/1 is. Should that

really be part of the definition of a scale?

on 8/17/04 4:37 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

>

>> But they do tell you what the notes are, once you know how the

>> representation works.

>

> No they don't. You need to specify how many Hertz 1/1 is. Should that

> really be part of the definition of a scale?

Well, obviously there's a scale and there's a scale. If I'm not worried

about subtleties of how absolute pitch affects perception of a scale then I

consider the scale to be the same object regardless of where I put 1/1.

That may be a matter of personal preference in how the term is used, and

perhaps I am technically incorrect, yet there seems to be a bit of precedent

for calling something that specifies relative pitches in cents variously

(loosely) a scale, a tuning, a temperament.

In any case the absolute pitch distinction is an orthogonal distinction from

the other distinctions made between temperament, tuning, scale, although I

imagine some would argue that only a scale has an absolute pitch reference.

So do we continue to be loose about scales that are relative versus

absolute, or do we need an adjective here?

-Kurt

on 8/17/04 1:00 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

>

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

>>>> As far as regular temperaments are concerned, one doesn't get to

>>>> tune the pitches independently. Tuning pitches independently gives

>>>> you what Gene would call a circulating temperament.

>>>

>>> Only if it circulates!

>>

>> So whaddya call an unequal scale which is playable in all keys

>> but which doesn't have all the good keys lumped together on the

>> chain of fifths?

>

> I call it circulating. I think someone (Parizek?) suggested that

> should be circular.

No I don't think he intended any distinction between circulating and

circular. He thought I might have intended such a distinction and I

corrected him.

Why would something that isn't at all round be called circular? If anything

"circulating" would be the less specific term if the two would be

distinguished, i.e. the answer to Carl's question should be "circulating"

and the name of the more specific kind of scale in which the good keys are

grouped together and there is a gradual progression should be called

"circular". But I'm not advocating making this distinction without knowing

a bit more about how the terms "circulating" and "circular" are currently

used in the literature in various languages. It would appear that

"circular" and "circulating" might need to be kept synonymous. Maybe some

speakers of non-english languages can confirm whether their word for

"circulating" is something they translate in english as "circular".

One of my own first temperament designs was a 12-tone well-temperered scale

in which the fifth size was the et fifth plus a sin function of the

circle-of-fifths position at some phase offset and some amplitude (I don't

have the details here). Therefore the third size variation was also a sin

function with a slightly different phase. Plotting the two against each

other around the circle of fifths would yield an ellipse. It occurs to me

that plotting the relation 3rd size vs 5th size around the circle of fifths

might in the case of many well-behaved circulating temperaments yield a

closed figure whose shape might reveal something interesting about a scale.

It might also be better to plot on non-orthogonal axes to make the result

tend to be more round rather than flattish.

-Kurt

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

wrote:

> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

>

> > But they do tell you what the notes are, once you know how the

> > representation works.

>

> No they don't. You need to specify how many Hertz 1/1 is.

You're right. But they do tell you what the scale is, which is

really the point.

> Should that

> really be part of the definition of a scale?

In general, no.

But occasionally the author of the scale says it is, and I think

this is sometimes reported in the comment at the start of the .scl

file.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

> wrote:

> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> >

> > > But they do tell you what the notes are, once you know how the

> > > representation works.

> >

> > No they don't. You need to specify how many Hertz 1/1 is.

>

> You're right. But they do tell you what the scale is, which is

> really the point.

Only if you don't define "scale" your way.

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

>

> /tuning/topicId_55471.html#55537

>

> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> > > if i had somebody around who could have given me

> > > answers like that 10 years ago, i would have saved

> > > myself a ridiculous amount of time and effort.

> >

> > Yes Monz,

> >

> > As I've said many times, I don't know what we'd do without your

> > encyclopedia. But don't you think that when an encyclopedia that's

> > edited by one person starts having entries for new terms added at

> > the whim of that editor and deleted again within a few days and

new

> > terms put in their place, it seriously undermines the authority of

> > that encyclopedia. Why should anyone take any notice of any of it

> if

> > that sort of thing can happen?

>

>

> ***With my apologies to my friend "the Monz," I intend to agree with

> Dave here. Monz is intrigued by "exotica"... see his historical

> early-civilization websites... Quite frankly, I think "exotica"

> should not be a part of terminology and rather than "colorful"

> creates an alternate club... a kind of tuning speakeasy (easyspeak??

)

>

> J. Pehrson

it's an encyclopaedia, so i will attempt to put

*everything* about tuning in it.

anyone who reads it is free to choose what they

read and what they don't read.

as i continue to learn more and more about

tuning-theory and history, it will also be added

into the software in future upgrades. and i'm

going to try as hard as i can to integrate

the Encyclopaedia and the software together.

hey, i'm an artist and the Encyclopaedia and

Musica are together my biggest and most important

work, and they'll both continue to evolve as long

as i'm able to work on them.

someone (probably Dave?) suggested that i mark

the usage of words: "obsolete", "archaic", etc.

that's a good idea ... it would take a while to

do that for all the pages which already exist,

but it's no problem to do it with new ones as i

create them.

i started to get that rolling by redoing the old

"um" and "vum" pages, so now, since those two

terms are archived quite heavily for a couple

of weeks worth of posts in this list, people can

look them up and be referred to their replacements.

that is much better than just having them disappear.

-monz

Hi, Monz,

on 8/18/04 12:18 PM, monz <monz@tonalsoft.com> wrote:

>>> As I've said many times, I don't know what we'd do without your

>>> encyclopedia. But don't you think that when an encyclopedia that's

>>> edited by one person starts having entries for new terms added at

>>> the whim of that editor and deleted again within a few days and

> new

>>> terms put in their place, it seriously undermines the authority of

>>> that encyclopedia. Why should anyone take any notice of any of it

>> if

>>> that sort of thing can happen?

>>

>> ***With my apologies to my friend "the Monz," I intend to agree with

>> Dave here. Monz is intrigued by "exotica"... see his historical

>> early-civilization websites... Quite frankly, I think "exotica"

>> should not be a part of terminology and rather than "colorful"

>> creates an alternate club... a kind of tuning speakeasy (easyspeak??

>

> it's an encyclopaedia, so i will attempt to put

> *everything* about tuning in it.

>

> anyone who reads it is free to choose what they

> read and what they don't read.

People inherently trust a source such as an encyclopedia (any source which

claims "authority") unless they learn to distrust it. So being free to

chose what to read is not very relevant to the "newbie" etc.

And so your following idea sounds good...

> someone (probably Dave?) suggested that i mark

> the usage of words: "obsolete", "archaic", etc.

> that's a good idea

and if you could extend the list to include "tentative", "experimental" etc.

that would take care of the issue.

In fact it might even be good to have an approximate date for each term, or

perhaps each definition in case a term has several definitions from

different centuries, etc.

-Kurt

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Hi, Monz,

>

> on 8/18/04 12:18 PM, monz <monz@t...> wrote:

>

> > someone (probably Dave?) suggested that i mark

> > the usage of words: "obsolete", "archaic", etc.

> > that's a good idea

>

> and if you could extend the list to include

> "tentative", "experimental" etc.

> that would take care of the issue.

yes, i'm going to start doing that kind of thing.

or at least explain with a sentence in the body of

the definition that the term was invented by so-and-so

at such-and-such date.

> In fact it might even be good to have an approximate date

> for each term, or perhaps each definition in case a term

> has several definitions from different centuries, etc.

there's one thing that i have always been very careful

to do: when i copy something from a tuning list or

tuning-math list post, i make sure to include the message

number and date information.

as the years have passed, i've gotten into the habit

of documenting everything in the Encyclopaedia better.

these days, when Paul helps me fix an erroneous definition,

i put "with Paul Erlich" in the update line at the bottom.

-monz